This document defines and provides examples of continuous functions between topological spaces. It can be summarized as follows:
1) A function f from a topological space X to a topological space Y is continuous if the preimage of every open set in Y is open in X.
2) Examples of continuous functions include identity functions, constant functions, and compositions of continuous functions.
3) A function from a space X to a product space Y×Z is continuous if and only if its coordinate projections to Y and Z are both continuous.
Some forms of N-closed Maps in supra Topological spacesIOSR Journals
In this paper, we introduce the concept of N-closed maps and we obtain the basic properties and
their relationships with other forms of N-closed maps in supra topological spaces.
The document contains proofs of various claims about continuous functions between metric spaces. It begins by proving that if a function f is continuous on closed subsets A and B of a metric space E whose union is E, then f is continuous on E (Problem 3). It then proves similar claims about continuity of nondecreasing functions between open intervals in R (Problem 4) and about a function's oscillation and continuity (Problem 5). The document proves several other properties of continuous functions.
This document discusses Urysohn's lemma, which states that a topological space is normal if any two disjoint closed subsets can be separated by a continuous function. It provides background on mathematician Pavel Urysohn, defines key terms like normal space, and outlines the proof of Urysohn's lemma, which constructs a continuous function separating two disjoint closed sets using dyadic fractions. Applications of the lemma include formulating other topological properties and solving extension theorems.
Let X bea topological space and let P be a family of disjoint nonempt.pdfinfo532468
Let X bea topological space and let P be a family of disjoint nonempty subsets of A siten P=X.
We say that P is a partition of X. We can form a new space Y, called an identification space, as
follows. The points of Y are the elements of P. The function :XY sends each point of X to the
subset of P containing it. We deem a subset U of Y to be open if and only if 1(U) is oper in X.
This is called the identification topology on Y. Any function f:XY toa set Y gives rise to a
partition of X whose members are the stbbets f1(y) where yY. Let Y. denote the identification
space associated with this partition, and :XY, the usual map. The exercises make reference to the
above notation and the following proofs. Proof A. The points of Y, are the sets {f1(y)} where
yY. Define h:Y,Y by h({f1(y)})=y. Then h is a bijection and satisfics h=f,h1f=. By one of the
theorems in these exercises, h is continuous, and h1 is continuous since we know that it is
continuous if tud ouly if the composition h1f:XY, is continuous. The result follows. Proof B. The
points of Y, are the sets {1(y)} where yY. Define h: YY by h((f1(y)f)=4. Then h is a bijection
and katisfies h=f,h1f=. By one of the theorems in these exercises, h is contimuons, and h1 is
continuons since the composite of continuous functions is continuots. The mesult follows. Proof
C. A closed subset of the compact space X is compuct and its image under the cont inuons
function f is therefore a compact subset of Y. But any compact subset of a Hausdorif space is
closed. Therefore f takes dosed sets to closed sets. We can now apply one of the theorems in
these excreises. Proof D. Let U be an open subeat of Z. Then f1(U) is open in Y if and only if
1(f1(U)) ) is upen in X. The result follown. Proof E Let U be asubet of Y for which f1(U) is open
in X. Let U=Y\U and note that f1(U)=X\f1(U) is closed in X. Since f is onto, we lanve
f(f1(U))=U, and therefore
Theorem Let f:XY be an onto map. If f maps closed sets of X to closed sets of Y then f is an
identification map. (A) The theorem is true because of Proof A above. (B) The theorem is true
because of Proof B above. (C) The theorem is true because of Proof C above. (D) The theorem is
true because of Proof D above. (E) The theorem is true because of Proof E above. (F) The
theorem is false. (G) The theorem is true but its truth is not established by any of the above
proofs. Click the button to the lef of the appropriate letter. Theorem. Let f:XY be an onto map. If
X is compact and Y is Hausdorff, then f is an identification map. (A) The theorem is true because
of Proof A above. (B) The theorem is true because of Proof B above. (C) The theorem is true
because of Proof C above. (D) The theorem is true because of Proof D above. (E) The theorem is
true becanse of Proof E above. (F) The theorem is fake. (G) The throrem is true but its truth is
not established by any of the ahowe prools. Click the button to the left of the approprinte letter..
This document presents research on extendable sets in the real numbers (R) and their application to the Lyapunov stability comparison principle of ordinary differential equations. It begins with definitions of the real numbers and extendable sets. It then reviews existing definitions of extension, including Urysohn's lemma and Tietze extension theorem. The main result proved is that every compact subset of R is extendable, while non-compact subsets are not. It concludes by extensively applying these results to prove important theorems regarding the comparison principle of Lyapunov stability theory in ordinary differential equations.
This document summarizes a research article that defines extendable sets in the real numbers (R) and applies this concept to proofs involving the Lyapunov stability comparison principle of ordinary differential equations. It begins with the author's own definition of an extension on R and a basic result called the basic extension fact for R. Existing definitions and theorems on extension, such as Urysohn's lemma and Tietze's extension theorem, are then reviewed. The document concludes by extensively applying these concepts to resolve proofs involving the comparison principle of Lyapunov stability theory.
Research Inventy : International Journal of Engineering and Scienceinventy
esearch Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Some forms of N-closed Maps in supra Topological spacesIOSR Journals
In this paper, we introduce the concept of N-closed maps and we obtain the basic properties and
their relationships with other forms of N-closed maps in supra topological spaces.
The document contains proofs of various claims about continuous functions between metric spaces. It begins by proving that if a function f is continuous on closed subsets A and B of a metric space E whose union is E, then f is continuous on E (Problem 3). It then proves similar claims about continuity of nondecreasing functions between open intervals in R (Problem 4) and about a function's oscillation and continuity (Problem 5). The document proves several other properties of continuous functions.
This document discusses Urysohn's lemma, which states that a topological space is normal if any two disjoint closed subsets can be separated by a continuous function. It provides background on mathematician Pavel Urysohn, defines key terms like normal space, and outlines the proof of Urysohn's lemma, which constructs a continuous function separating two disjoint closed sets using dyadic fractions. Applications of the lemma include formulating other topological properties and solving extension theorems.
Let X bea topological space and let P be a family of disjoint nonempt.pdfinfo532468
Let X bea topological space and let P be a family of disjoint nonempty subsets of A siten P=X.
We say that P is a partition of X. We can form a new space Y, called an identification space, as
follows. The points of Y are the elements of P. The function :XY sends each point of X to the
subset of P containing it. We deem a subset U of Y to be open if and only if 1(U) is oper in X.
This is called the identification topology on Y. Any function f:XY toa set Y gives rise to a
partition of X whose members are the stbbets f1(y) where yY. Let Y. denote the identification
space associated with this partition, and :XY, the usual map. The exercises make reference to the
above notation and the following proofs. Proof A. The points of Y, are the sets {f1(y)} where
yY. Define h:Y,Y by h({f1(y)})=y. Then h is a bijection and satisfics h=f,h1f=. By one of the
theorems in these exercises, h is continuous, and h1 is continuous since we know that it is
continuous if tud ouly if the composition h1f:XY, is continuous. The result follows. Proof B. The
points of Y, are the sets {1(y)} where yY. Define h: YY by h((f1(y)f)=4. Then h is a bijection
and katisfies h=f,h1f=. By one of the theorems in these exercises, h is contimuons, and h1 is
continuons since the composite of continuous functions is continuots. The mesult follows. Proof
C. A closed subset of the compact space X is compuct and its image under the cont inuons
function f is therefore a compact subset of Y. But any compact subset of a Hausdorif space is
closed. Therefore f takes dosed sets to closed sets. We can now apply one of the theorems in
these excreises. Proof D. Let U be an open subeat of Z. Then f1(U) is open in Y if and only if
1(f1(U)) ) is upen in X. The result follown. Proof E Let U be asubet of Y for which f1(U) is open
in X. Let U=Y\U and note that f1(U)=X\f1(U) is closed in X. Since f is onto, we lanve
f(f1(U))=U, and therefore
Theorem Let f:XY be an onto map. If f maps closed sets of X to closed sets of Y then f is an
identification map. (A) The theorem is true because of Proof A above. (B) The theorem is true
because of Proof B above. (C) The theorem is true because of Proof C above. (D) The theorem is
true because of Proof D above. (E) The theorem is true because of Proof E above. (F) The
theorem is false. (G) The theorem is true but its truth is not established by any of the above
proofs. Click the button to the lef of the appropriate letter. Theorem. Let f:XY be an onto map. If
X is compact and Y is Hausdorff, then f is an identification map. (A) The theorem is true because
of Proof A above. (B) The theorem is true because of Proof B above. (C) The theorem is true
because of Proof C above. (D) The theorem is true because of Proof D above. (E) The theorem is
true becanse of Proof E above. (F) The theorem is fake. (G) The throrem is true but its truth is
not established by any of the ahowe prools. Click the button to the left of the approprinte letter..
This document presents research on extendable sets in the real numbers (R) and their application to the Lyapunov stability comparison principle of ordinary differential equations. It begins with definitions of the real numbers and extendable sets. It then reviews existing definitions of extension, including Urysohn's lemma and Tietze extension theorem. The main result proved is that every compact subset of R is extendable, while non-compact subsets are not. It concludes by extensively applying these results to prove important theorems regarding the comparison principle of Lyapunov stability theory in ordinary differential equations.
This document summarizes a research article that defines extendable sets in the real numbers (R) and applies this concept to proofs involving the Lyapunov stability comparison principle of ordinary differential equations. It begins with the author's own definition of an extension on R and a basic result called the basic extension fact for R. Existing definitions and theorems on extension, such as Urysohn's lemma and Tietze's extension theorem, are then reviewed. The document concludes by extensively applying these concepts to resolve proofs involving the comparison principle of Lyapunov stability theory.
Research Inventy : International Journal of Engineering and Scienceinventy
esearch Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
This document introduces the concept of fuzzy compact-open topology. Some key points:
- The fuzzy compact-open topology is defined on the class of fuzzy continuous functions between two fuzzy topological spaces.
- An evaluation map from the product of this function class and the domain space into the range space is shown to be fuzzy continuous.
- For fuzzy locally compact Hausdorff domain and range spaces, there is an exponential law isomorphism between the function class with the compact-open topology and the product of this class with the domain space.
The authors Selvi.R, Thangavelu.P and
Anitha.m introduced the concept of
-continuity between a
topological space and a non empty set where
{L, M, R, S}
[4]. Navpreet singh Noorie and Rajni Bala[3] introduced the
concept of f#
function to characterize the closed, open and
continuous functions. In this paper, the concept of Semi- -
continuity is introduced and its properties are investigated and
Semi- -continuity is further characterized by using f#
functions
On Extendable Sets in the Reals (R) With Application to the Lyapunov Stabilit...BRNSS Publication Hub
This document summarizes a research article that defines extendable sets in the real numbers (R) and applies this concept to the Lyapunov stability comparison principle of ordinary differential equations. It begins with the author's own definition of extension on R and a basic result called the basic extension fact for R. It then reviews existing definitions and theorems on extension, including Urysohn's lemma and Tietze's extension theorem. The document concludes by extensively applying these results to prove some important results relating to the comparison principle of Lyapunov stability theory in ordinary differential equations.
This document provides an overview of Tychonoff's theorem in topology. It begins with definitions of filters, ultrafilters, and convergence. It then discusses compactness and proves that a space is compact if and only if every ultrafilter converges. Finally, it states and proves Tychonoff's theorem, which says the product of topological spaces is compact if and only if each factor space is compact. The proof uses previous results about filters, ultrafilters, and convergence in product spaces.
On Analytic Review of Hahn–Banach Extension Results with Some GeneralizationsBRNSS Publication Hub
The useful Hahn–Banach theorem in functional analysis has significantly been in use for many years ago. At this point in time, we discover that its domain and range of existence can be extended point wisely so as to secure a wider range of extendibility. In achieving this, we initially reviewed the existing traditional Hahn–Banach extension theorem, before we carefully and successfully used it to generate the finite extension form as in main results of section three.
In the present paper , we introduce and study the concept of gr- Ti- space (for i =0,1,2) and
obtain the characterization of gr –regular space , gr- normal space by using the notion of gr-open
sets. Further, some of their properties and results are discussed.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
Prove that compactness is a topological property. SolutionSupp.pdffasttracktreding
Prove that compactness is a topological property.
Solution
Suppose that X and Y are topological spaces, and f : X Y is a homeomorphism. That is, f : X Y
is a bijection, f and f 1 : Y X are continuous.
(i) Compactness is a topological property:
This is to show that if X is compact, then Y is also.
Suppose that X is compact. We want to show that Y is also compact.
Our strategy is to apply the definition, and show that every infinite sequence of points in Y has a
limit point in Y . Let {yi} i=1 be an infinite sequence of points in Y .
Let xi = f 1 (yi), i = 1, 2, ... Then {xi}i=1 is an infinite sequence of points in X.
Since X is compact, {xi}i=1 has a limit point a in X.
Since f : X Y is continuous, f(a) is a limit point of {f(xi)}i=1 = {yi}i =1.
Since f(a) Y , the sequence {yi}i=1 has a limit point in Y .
By the definition of compact sets, Y is compact..
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document provides an overview and proofs of several theorems related to the Hahn-Banach theorem. It begins with an introduction to linear functionals and the Hahn-Banach theorem. It then presents two main theorems - the Hahn-Banach theorem and the topological Hahn-Banach theorem. The document provides proofs of these theorems and several related theorems using the Hahn-Banach extension lemma. It also discusses consequences of the Hahn-Banach extension form and provides proofs of the theorems using the lemma.
On Some Continuous and Irresolute Maps In Ideal Topological Spacesiosrjce
In this paper we introduce some continuous and irresolute maps called
δ
ˆ
-continuity,
δ
ˆ
-irresolute,
δ
ˆ
s-continuity and
δ
ˆ
s-irresolute maps in ideal topological spaces and study some of their properties.
The document discusses one-to-one functions and inverse functions. A function f(x) is one-to-one if different inputs produce different outputs. The inverse of a one-to-one function f(x), denoted f^-1(x), is a function with the domain and range swapped such that f^-1(f(x)) = x and f(f^-1(x)) = x. While some inverse functions can be solved for algebraically, in general inverse functions must be determined through other means as algebraic solutions are not always possible.
This document discusses unbounded transcendent formal power series over a finite field Fq((X-1)). It presents a new transcendence criterion for continued fractions in this field. Specifically, it constructs a family of transcendental continued fractions with unbounded partial quotients obtained from algebraic elements. The main result proves that if a formal power series can be approximated by a family of algebraic series with increasingly long blocks in their continued fraction expansions, then the formal power series must be transcendental. An example is also given to illustrate the main result.
This document discusses unbounded transcendent formal power series over a finite field Fq((X-1)). It presents a new general result that establishes a transcendence criterion for continued fractions with unbounded partial quotients constructed from algebraic elements. Specifically, the theorem shows that if a formal power series can be approximated by a family of algebraic series with increasing block lengths, then the formal power series is transcendental. The proof uses previous results on continued fractions over finite fields and algebraic degree estimates. An example is also given to illustrate the main result.
Some Results on Fuzzy Supra Topological SpacesIJERA Editor
This document discusses fuzzy supra topological spaces. It begins by introducing fuzzy sets and fuzzy topology. It then defines fuzzy supra topological spaces and related concepts such as fuzzy supra open sets, fuzzy supra neighborhoods, and fuzzy supra continuity. It presents some preliminary definitions and propositions. It then obtains some results on fuzzy supra topological spaces, including that a continuous function between fuzzy supra topological spaces has a fuzzy supra closed graph if the range space is Hausdorff, and that a continuous injection with a fuzzy supra closed graph implies the domain space is Hausdorff.
MA500-2: Topological Structures 2016
Aisling McCluskey, Daron Anderson
[email protected], [email protected]
Contents
0 Preliminaries 2
1 Topological Groups 8
2 Morphisms and Isomorphisms 15
3 The Second Isomorphism Theorem 27
4 Topological Vector Spaces 42
5 The Cayley-Hamilton Theorem 43
6 The Arzelà-Ascoli theorem 44
7 Tychonoff ’s Theorem if Time Permits 45
Continuous assessment 30%; final examination 70%. There will be a weekly
workshop led by Daron during which there will be an opportunity to boost
continuous assessment marks based upon workshop participation as outlined in
class.
This module is self-contained; the notes provided shall form the module text.
Due to the broad range of topics introduced, there is no recommended text.
However General Topology by R. Engelking is a graduate-level text which has
relevant sections within it. Also Undergraduate Topology: a working textbook by
McCluskey and McMaster is a useful revision text. As usual, in-class discussion
will supplement the formal notes.
1
0 PRELIMINARIES
0 Preliminaries
Reminder 0.1. A topology τ on the set X is a family of subsets of X, called
the τ-open sets, satisfying the three axioms.
(1) Both sets X and ∅ are τ-open
(2) The union of any subfamily is again a τ-open set
(3) The intersection of any two τ-open sets is again a τ-open set
We refer to (X,τ) as a topological space. Where there is no danger of ambi-
guity, we suppress reference to the symbol denoting the topology (in this case,
τ) and simply refer to X as a topological space and to the elements of τ as its
open sets. By a closed set we mean one whose complement is open.
Reminder 0.2. A metric on the set X is a function d: X×X → R satisfying
the five axioms.
(1) d(x,y) ≥ 0 for all x,y ∈ X
(2) d(x,y) = d(y,x) for x,y ∈ X
(3) d(x,x) = 0 for every x ∈ X
(4) d(x,y) = 0 implies x = y
(5) d(x,z) ≤ d(x,y) + d(y,z) for all x,y,z ∈ X
Axiom (5) is often called the triangle inequality.
Definition 0.3. If d′ : X × X → R satisfies axioms (1), (2), (3) and (5) but
maybe not (4) then we call it a pseudo-metric.
Reminder 0.4. Every metric on X induces a topology on X, called the metric
topology. We define an open ball to be a set of the form
B(x,r) = {y ∈ X : d(x,y) < r}
for any x ∈ X and r > 0. Then a subset G of X is defined to be open (wrt the
metric topology) if for each x ∈ G, there is r > 0 such that B(x,r) ⊂ G. Thus
open sets are arbitrary unions of open balls.
Topological Structures 2016 2 Version 0.15
0 PRELIMINARIES
The definition of the metric topology makes just as much sense when we are
working with a pseudo-metric. Open balls are defined in the same manner, and
the open sets are exactly the unions of open balls. Pseudo-metric topologies are
often neglected because they do not have the nice property of being Hausdorff.
Reminder 0.5. Suppose f : X → Y is a function between the topological
spaces X and Y . We say f is continuous to mean that whenever U is open in
Y ...
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
Fibrewise near compact and locally near compact spacesAlexander Decker
This document defines and studies new concepts of fibrewise topological spaces over a base set B, namely fibrewise near compact and fibrewise locally near compact spaces. These are generalizations of near compact and locally near compact topological spaces. Key definitions include:
1. Fibrewise near compact spaces, where the projection map is a "near proper" function and each fibre is near compact.
2. Fibrewise locally near compact spaces, where each point has a neighbourhood whose closure is near compact.
3. Relationships between these concepts and some fibrewise near separation axioms are also studied.
This document introduces the concept of fuzzy compact-open topology. Some key points:
- The fuzzy compact-open topology is defined on the class of fuzzy continuous functions between two fuzzy topological spaces.
- An evaluation map from the product of this function class and the domain space into the range space is shown to be fuzzy continuous.
- For fuzzy locally compact Hausdorff domain and range spaces, there is an exponential law isomorphism between the function class with the compact-open topology and the product of this class with the domain space.
The authors Selvi.R, Thangavelu.P and
Anitha.m introduced the concept of
-continuity between a
topological space and a non empty set where
{L, M, R, S}
[4]. Navpreet singh Noorie and Rajni Bala[3] introduced the
concept of f#
function to characterize the closed, open and
continuous functions. In this paper, the concept of Semi- -
continuity is introduced and its properties are investigated and
Semi- -continuity is further characterized by using f#
functions
On Extendable Sets in the Reals (R) With Application to the Lyapunov Stabilit...BRNSS Publication Hub
This document summarizes a research article that defines extendable sets in the real numbers (R) and applies this concept to the Lyapunov stability comparison principle of ordinary differential equations. It begins with the author's own definition of extension on R and a basic result called the basic extension fact for R. It then reviews existing definitions and theorems on extension, including Urysohn's lemma and Tietze's extension theorem. The document concludes by extensively applying these results to prove some important results relating to the comparison principle of Lyapunov stability theory in ordinary differential equations.
This document provides an overview of Tychonoff's theorem in topology. It begins with definitions of filters, ultrafilters, and convergence. It then discusses compactness and proves that a space is compact if and only if every ultrafilter converges. Finally, it states and proves Tychonoff's theorem, which says the product of topological spaces is compact if and only if each factor space is compact. The proof uses previous results about filters, ultrafilters, and convergence in product spaces.
On Analytic Review of Hahn–Banach Extension Results with Some GeneralizationsBRNSS Publication Hub
The useful Hahn–Banach theorem in functional analysis has significantly been in use for many years ago. At this point in time, we discover that its domain and range of existence can be extended point wisely so as to secure a wider range of extendibility. In achieving this, we initially reviewed the existing traditional Hahn–Banach extension theorem, before we carefully and successfully used it to generate the finite extension form as in main results of section three.
In the present paper , we introduce and study the concept of gr- Ti- space (for i =0,1,2) and
obtain the characterization of gr –regular space , gr- normal space by using the notion of gr-open
sets. Further, some of their properties and results are discussed.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
Prove that compactness is a topological property. SolutionSupp.pdffasttracktreding
Prove that compactness is a topological property.
Solution
Suppose that X and Y are topological spaces, and f : X Y is a homeomorphism. That is, f : X Y
is a bijection, f and f 1 : Y X are continuous.
(i) Compactness is a topological property:
This is to show that if X is compact, then Y is also.
Suppose that X is compact. We want to show that Y is also compact.
Our strategy is to apply the definition, and show that every infinite sequence of points in Y has a
limit point in Y . Let {yi} i=1 be an infinite sequence of points in Y .
Let xi = f 1 (yi), i = 1, 2, ... Then {xi}i=1 is an infinite sequence of points in X.
Since X is compact, {xi}i=1 has a limit point a in X.
Since f : X Y is continuous, f(a) is a limit point of {f(xi)}i=1 = {yi}i =1.
Since f(a) Y , the sequence {yi}i=1 has a limit point in Y .
By the definition of compact sets, Y is compact..
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document provides an overview and proofs of several theorems related to the Hahn-Banach theorem. It begins with an introduction to linear functionals and the Hahn-Banach theorem. It then presents two main theorems - the Hahn-Banach theorem and the topological Hahn-Banach theorem. The document provides proofs of these theorems and several related theorems using the Hahn-Banach extension lemma. It also discusses consequences of the Hahn-Banach extension form and provides proofs of the theorems using the lemma.
On Some Continuous and Irresolute Maps In Ideal Topological Spacesiosrjce
In this paper we introduce some continuous and irresolute maps called
δ
ˆ
-continuity,
δ
ˆ
-irresolute,
δ
ˆ
s-continuity and
δ
ˆ
s-irresolute maps in ideal topological spaces and study some of their properties.
The document discusses one-to-one functions and inverse functions. A function f(x) is one-to-one if different inputs produce different outputs. The inverse of a one-to-one function f(x), denoted f^-1(x), is a function with the domain and range swapped such that f^-1(f(x)) = x and f(f^-1(x)) = x. While some inverse functions can be solved for algebraically, in general inverse functions must be determined through other means as algebraic solutions are not always possible.
This document discusses unbounded transcendent formal power series over a finite field Fq((X-1)). It presents a new transcendence criterion for continued fractions in this field. Specifically, it constructs a family of transcendental continued fractions with unbounded partial quotients obtained from algebraic elements. The main result proves that if a formal power series can be approximated by a family of algebraic series with increasingly long blocks in their continued fraction expansions, then the formal power series must be transcendental. An example is also given to illustrate the main result.
This document discusses unbounded transcendent formal power series over a finite field Fq((X-1)). It presents a new general result that establishes a transcendence criterion for continued fractions with unbounded partial quotients constructed from algebraic elements. Specifically, the theorem shows that if a formal power series can be approximated by a family of algebraic series with increasing block lengths, then the formal power series is transcendental. The proof uses previous results on continued fractions over finite fields and algebraic degree estimates. An example is also given to illustrate the main result.
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This document discusses fuzzy supra topological spaces. It begins by introducing fuzzy sets and fuzzy topology. It then defines fuzzy supra topological spaces and related concepts such as fuzzy supra open sets, fuzzy supra neighborhoods, and fuzzy supra continuity. It presents some preliminary definitions and propositions. It then obtains some results on fuzzy supra topological spaces, including that a continuous function between fuzzy supra topological spaces has a fuzzy supra closed graph if the range space is Hausdorff, and that a continuous injection with a fuzzy supra closed graph implies the domain space is Hausdorff.
MA500-2: Topological Structures 2016
Aisling McCluskey, Daron Anderson
[email protected], [email protected]
Contents
0 Preliminaries 2
1 Topological Groups 8
2 Morphisms and Isomorphisms 15
3 The Second Isomorphism Theorem 27
4 Topological Vector Spaces 42
5 The Cayley-Hamilton Theorem 43
6 The Arzelà-Ascoli theorem 44
7 Tychonoff ’s Theorem if Time Permits 45
Continuous assessment 30%; final examination 70%. There will be a weekly
workshop led by Daron during which there will be an opportunity to boost
continuous assessment marks based upon workshop participation as outlined in
class.
This module is self-contained; the notes provided shall form the module text.
Due to the broad range of topics introduced, there is no recommended text.
However General Topology by R. Engelking is a graduate-level text which has
relevant sections within it. Also Undergraduate Topology: a working textbook by
McCluskey and McMaster is a useful revision text. As usual, in-class discussion
will supplement the formal notes.
1
0 PRELIMINARIES
0 Preliminaries
Reminder 0.1. A topology τ on the set X is a family of subsets of X, called
the τ-open sets, satisfying the three axioms.
(1) Both sets X and ∅ are τ-open
(2) The union of any subfamily is again a τ-open set
(3) The intersection of any two τ-open sets is again a τ-open set
We refer to (X,τ) as a topological space. Where there is no danger of ambi-
guity, we suppress reference to the symbol denoting the topology (in this case,
τ) and simply refer to X as a topological space and to the elements of τ as its
open sets. By a closed set we mean one whose complement is open.
Reminder 0.2. A metric on the set X is a function d: X×X → R satisfying
the five axioms.
(1) d(x,y) ≥ 0 for all x,y ∈ X
(2) d(x,y) = d(y,x) for x,y ∈ X
(3) d(x,x) = 0 for every x ∈ X
(4) d(x,y) = 0 implies x = y
(5) d(x,z) ≤ d(x,y) + d(y,z) for all x,y,z ∈ X
Axiom (5) is often called the triangle inequality.
Definition 0.3. If d′ : X × X → R satisfies axioms (1), (2), (3) and (5) but
maybe not (4) then we call it a pseudo-metric.
Reminder 0.4. Every metric on X induces a topology on X, called the metric
topology. We define an open ball to be a set of the form
B(x,r) = {y ∈ X : d(x,y) < r}
for any x ∈ X and r > 0. Then a subset G of X is defined to be open (wrt the
metric topology) if for each x ∈ G, there is r > 0 such that B(x,r) ⊂ G. Thus
open sets are arbitrary unions of open balls.
Topological Structures 2016 2 Version 0.15
0 PRELIMINARIES
The definition of the metric topology makes just as much sense when we are
working with a pseudo-metric. Open balls are defined in the same manner, and
the open sets are exactly the unions of open balls. Pseudo-metric topologies are
often neglected because they do not have the nice property of being Hausdorff.
Reminder 0.5. Suppose f : X → Y is a function between the topological
spaces X and Y . We say f is continuous to mean that whenever U is open in
Y ...
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1. UNIT – II
CONTINUOUS FUNCTION
Definition: Continuous Function X f Y
Let X and Y be topological spaces. V
A function f : X Y is said to be continuous if for f-1(V)
each open subsetV of Y, the set f-1 (V) is an open subset of X. Open
Note 1:
Recall that f-1 (V) is the set of all points x of X for which f(x) V. It is
empty if V does not intersect the image set f(X) of f.
Note 2:
Continuity of a function depends not only upon the function itself, but also
on the topologies specified for its domain and range. If we wish to emphasize
this fact, we can say that f is continuous relative to specified topologies on X
and Y.
Example-1:
Let us consider the function like those studied in analysis “ A real valued
function of a real variable”.
f : ℛ ℛ
i.e., f(x) = x
Example-2:
Let ℛ denote the set of real numbers in its usual topology and let ℛℓ
denote the same set in the lowerlimit topology.
Let f : ℛ ℛℓ be the identity function.
f(x) = x for every real number X. Then f is not a continuous function. The
inverse image of an open set [a, b) of ℛℓ equals itself which is not open in ℛ.
On the otherhand, the identity function g : ℛℓ ℛ is continuous, because
the inverse image of (a, b) is itself, which is open in ℛℓ .
Definition : Homeomorphism
Let X and Y be a topological spaces. Let f : X Y be a bijection.
2. If both f and f-1
are continuous, then f is called a Homeomorphism.
Theorem:
Statement:
Let X and Y be a topological spaces. Let f: X Y. Then the following
are equivalent.
(i ) f is continuous
(ii) for every subsetA of X, f(Ā ) f(A)
(iii) for every closedsetB of Y the setf-1
(B) is closedin X
(iv) for eachx X and eachneighbourhood V of f(x) there is a
neighbourhood U of x such that f(U) V
If the conclusion in (4) holds for the point x of X we say that f is
continuous at the point x.
Proof:
Let X and Y be the topological spaces. Let f : X Y.
(i) ⇒ (ii)
Assume that f is continuous. Let A be a subsetof X.
To prove: f(Ā) 𝒇(𝐀)
̅̅̅̅̅
Let x Ā. Then f(x) f(Ā)
if f(x) f(Ā) then we have to show that
f(x) 𝒇(𝐀)
̅̅̅̅̅
Let V be a neighbourhood of f(x). Then f-1 (V) is an open set of X
containing x. (∵ f is continuous )
Here x Ā and f-1 (V) is open.
∴ f-1 (V) must intersect A in some point y.
Then V intersects f(A) in the point f(y).
i.e., f(y) V f(A)
⇒V f(A) is non empty.
⇒ f(x) f(A)
∴ f(𝐴̅) f(A)
(ii) ⇒ (iii)
Let B closed in Y and let A = f-1 (B)
To prove: A is closed in X.
We have A Ā
3. if we prove Ā A then A = Ā
⇒ A is closed.
Let us prove : Ā A
Here A = f-1 (B) ⇒ f(A) B.
Let x Ā then f(x) f(Ā)
𝑓(𝐴)
̅̅̅̅̅̅
B = B since B is closed
i.e., f(x) B (or) x f-1 (B) = A
⇒ Ā A
Hence A = Ā
∴ A = f-1 (B) is closed.
(iii) ⇒ (i)
Let V be an open set in Y. Let B = Y-V, then B is closed in Y.
∴ f-1 (B) is closed in X. (by (iii)) Y V
f-1 (V) = f-1 (Y-B) f
= f-1 (Y) - f-1 (B) f-1 (V) f(y) = V
= X - f-1 (B)
∴ f-1 (V) is open.
Hence f is continuous.
(i) ⇒(iv)
Let x X and V be a neighbourhood of f(x). Then since f is continuous
f-1 (V) is a neighbourhood of x. Let f-1 (V) = U.
Then f(U) = f(f-1 (V)) V
For given x X and a neighbourhood V of f(x), there exist a
neighbourhood U of x such that f(U) V.
(iv)⇒(i)
Let V be an open set of Y.
Let x f-1 (V) then f(x) V.
By hypothesis,
∃ a neighbourhood Ux of x such that f(Ux) V
Then, Ux f-1 (f(Ux)) f-1 (V).
Hence f-1 (V) = ∪ Ux.
since each Ux is open and union of open sets is open, f-1 (V) is open in X.
4. Therefore f is continuous.
Hence the theorem.
FootNote:
i) A is always contained in f-1 (f(A))
i.e A f-1 (f(A))
ii) f (f-1(B)) B.
Result-1:
If the inverse image of every basis element is open, then f is continuous.
Proof:
Let f : XY and the inverse image of every basis element be open
Let V Y be a open in Y.
Then V= ∪ B𝛼
𝛼
f-1 (V) f-1 ( ∪ B𝛼 ) = ∪ f-1 (B𝛼)
𝛼 𝛼
⇒ f-1 (V) is open in X, since each f-1 (B𝛼) is open in X.
Definition: Open map
A map f : X Y is said to be an open map if for every open set U of X,
f(U) is open in Y.
Note:
Let f : X Y, then the map f-1 : Y X the inverse image of U under the
map f-1 is same as the image of U under the map f.
5. The homeomorphism can be defined as a bijective correspondence
f : X Y such that f(U) is open iff U is open.
Definition : Topologicalproperty
Let f : X Y be a homeomorphism. Any property of X that is entirely
expressed in terms of the topology of X yields, through the correspondencef, the
correspondingpropertyforthe spaceY, sucha propertyof Xis called topological
property.
Constructing Continuous Functions:
Theorem: Rules for constructing continuous function
Let X, Y and Z be topological spaces.
(a) Constantfunction:
If f : X Y maps all of X into the single point y0 of Y.
Then f is continuous.
Proof: X Y
F
> f-1 (V ) C
X C > f-1 (V ) C
Let V be an open set in Y.
Then f-1 (V ) = y0 V
X y0 V
∴ f-1 (V ) is open in X. (since both and X are open in X)
6. (b) RestrictionFunction:
Let A be a subspaceof X. Then the restriction function, restricting the
domain f /A : A Y is continuous.
Proof:
Let V be open in Y.Then f-1 (V ) is open in X, since f is continuous.
f-1 (V ) A is open in A and (f(A)-1 (V) = f-1 (V ) A
Thus (f(A))-1 (V) is open in A.
∴ f /A : A Y is continuous.
(c) Inclusion Function:
If A is a subspaceofX the inclusion function j : A X is continuous.
Proof:
Let U be open in X then j-1 (U) = A U is open in A.
(in subspacetopology)
∴ j-1 (U) is open in A.
i.e j : A X is continuous.
(d) Composition of continuous functions is continuous
If f : X Y and g: Y Z are continuous then the map gof : X Z
is continuous.
Proof:
Let V be an open set of Z then g-1 (V) is open in Y. (∵ g is continuous)
Since f is continuous f-1 (g-1 (V)) is open in X.
And f-1 (g-1 (V)) = (gof)-1 (V)
∴ for every open set V of Z, (gof)-1 (V) is open in X.
Hence gof : X Z is continuous.
(e) Restricting (or) Expanding the range:
(i) Let f : X Y be continuous. If Z is a subspaceof Y containing the
image set f(X), then the function g: X Z obtained by restricting the range of f
is continuous.
(ii) If Z is a spacehaving Y as a subspacethen the function fn , h : X Z
obtained by expanding the range of f is continuous.
7. Proof:
(i) Let f : X Y be continuous.
If Z is a subspaceof Y containing f(X)
We have f(X) Z Y.
Let g: X Z be a function.
To prove: g is continuous.
Let B be open in Z.
To prove:
g-1(B) is open in X.
Since B is open in the subspacetopology, B = U Z , where U is open in Y.
g-1(B) = f-1(B) = f-1 (U Z)
= f-1 (U) f-1 (Z)
= f-1 (U) X.
g-1(B) = f-1(U) ………..()
Since U is open in Y and f is continuous, X Z
f-1(U) is open in X. f Y
i.e., g-1(B) is open in X (by ()) >
Hence g : X Z is continuous. foj
(ii) Let Z contains Y as a subspace
given that f : X Y is continuous.
The inclusion function j: YZ is also continuous.
∴ Their composition (jof) : X Z is continuous.
i.e., The map h : X Z is continuous.
(f) Localformulation of Continuity:
The map f : X Y is continuous if X can be written as union of open
sets U𝛼 such that f / U𝛼 is continuous for each 𝛼.
Proof:
Let f : X Y and let X = U𝛼
Given that :
f U𝛼 : U𝛼 Y is continuous, for each 𝛼.
To prove: X Y
8. f U𝛼 : X Y is continuous.
Let V be open in Y.
claim:
f-1(V) is open in X.
Since f / U: U𝛼 Y is continuous, and V is open in Y.
(f/U𝛼)-1(V) is open in U𝛼.
(f/U𝛼)-1(V) = f-1 (V) U𝛼 is open in U𝛼.
Since U𝛼 is open in X, we have
f-1 (V) U𝛼 is open in X.
Now
arbitrary union of open sets is open.
⇒ f-1 (V) is open in X, Since f : X Y is continuous.
Hence proved.
Pasting Lemma:
Statement:
Let X = A⋃B where A and B are closed in X. Let f : A Y and g: B Y
be continuous. If f(x) = g(x) ∀ x AB then f & g combine to give a
continuous function.
h : X Y defined by setting
h(x) = f(x) , if xA
g(x) , if xB
Proof:
Let V be a closed set in Y. Then
h-1 (V) = f-1 (V) ⋃ g-1 (V) X f Y
Since f is continuous, A f-1(V)
f-1 (V) is closed in A and A closed in X AB
⇒ f-1 (V) is closed in X. B g V
g-1 (V) is closed in B and B closed in X
⇒ g-1 (V) is closed in X.
Since union of two closed sets is closed,
f-1 (V) ⋃ g-1 (V) is closed in X.
i.e., h-1 (V) is closed in X.
Therefore h is continuous.
9. Hence the proof.
Example 1:
For Pasting Lemma g(x)
Define h : ℛ ℛ by x ≤ 0
h(x) = x if x ≤ 0 x ≥ 0
x/2 if x ≥ 0 f(x)
f(x) = x, g(x) = x/2
A = { x: x ≤ 0 } = negative reals ⋃ {0} is closed.
B = { x: x ≥ 0 } = R+ ⋃ {0} is closed.
and R = A ⋃ B
A B = {0}
f(0) = 0, g(0) = 0.
Hence f(0) =g(0).
Hence by Pasting Lemma, h is continuous.
Example 2:
The pieces of the function must agree on the overlapping part of their
domains in Pasting Lemma. If not the function need not be continuous.
Let h1: ℛ ℛ defined by Y
h1(x) = x-2 if x ≤ 0 (0,2) g(x)
x+2 if x ≥ 0 X
x ≤ 0 x ≥ 0
f(x) = x-2, g(x) = x+2
A = {x: x ≤ 0} (0,-2)
= R - ⋃ {0} is closed.
B = {x: x ≥ 0 }
= R+ ⋃ {0} is closed.
ℛ = A⋃B.
A B = {0}
f(0) = -2 ≠ g(0) = 2.
From the graph it is clear that h1 is not continuous.
Example 3:
Let : ℛ ℛ
Let 𝓁(x) = x-2 if x < 0 , x+2 if x ≥ 0
10. A = {x: x < 0}
= R – is not closed.
We define a function 𝓁 mapping ℛ into ℛ and both the pieces are
continuous.
But 𝓁 is not continuous, the inverse image of the open set (1,3) is
non-open set [0,1).
Theorem2.4:
Maps Into Products: X XY
Statement:
Let f : A XY be given by the equation (f1(a),
f(a) = (f1(a), f2(a)) a f2(a))
Then f is continuous if and only if the function
f1 : A X and f2 : A Y are continuous.
The maps f1 and f2 are called the coordinate functions of f.
Proof: XY 1 X
Let 1 : XY X and U
2 : XY Y be projections onto
the first and second factors, respectively.
These maps are continuous. 1
-1
(U) = UY
For 1
-1
(U) = UY and XY X
2
-1
(U) = XV and these sets are open. 2 V
if U and V are open. Note that for each aA.
f1(a) = 1(f(a)) and 2
-1
(V)=XxV
f2(a) = 2(f(a)).
If the function f is continuous then f1 and f2 are composites ofcontinuous
function and therefore continuous.
Conversely, A XY X
Supposethat f1 and f2 are continuous. > f 1>
We show that for each basis element UV for a f(a) 1f(a)
the topology of XY,
its inverse image f-1 (UV) is open. <
A point a is in f-1 (UV) iff f(a) UV. f1
i.e., iff f1(a) U and f2(a) V.
f-1 (UV) = f1
-1 (U ) f2
-1 (V) X XY
Since both of the sets f1
-1 (U ) and f2
-1 (V) are open, f
11. so is their intersection.
Hence the proof.
Definition : Limit of the Sequence
If the sequence {xn} of points of the Hausdorff Space X converges to a
point x of X. We write xn x and call x as a limit of the sequence {xn}.
The Product Topology:
Definition: J-tuple
Let J be an indexed set given a set X. We define a J-tuple of
elements of X to be the function
X: J X if 𝛼 is an element of J. We denote the value if X at 𝛼 by
(X(a)=) x𝛼 rather than x(𝛼).
Then x𝛼 is called the 𝛼th co-ordinate of X.
The function X itself is denoted by the symbol (x𝛼) 𝛼J . We denote the set
of all J-tuples of elements of X by XJ.
Definition:
Let {A𝛼}𝛼J be an indexed family of sets. Let
The Cartesian productof this indexed family denoted by
is defined to be the set of all J-tuples (x𝛼)𝛼J elements of X such that
x𝛼 A𝛼 for each 𝛼 J.
i.e., it is the set of all functions
such that x(𝛼) A𝛼 for each 𝛼.
Note:
If all the sets A𝛼 are equal to X, then the cartesian product
𝛼J A𝛼 is just the set XJ of J-tuples of elements of X.
12. Definition: Box Topology
Let {X𝛼}𝛼J be an indexed family of topological spaces. Thenthe basis for
a topology on the productspace 𝛼J X𝛼 is the collection of all sets of the
form 𝛼J U𝛼 where U𝛼 is open in X for each 𝛼 J.
The topology generated by this basis is called the Box topology.
Note:
The collection satisfies the first condition for a basis because X𝛼 is itself
a basis element and it satisfies the 2nd
condition because the intersection of any
two basis element is another basis element.
Definition: ProjectionMapping
Let be the function assigning to each element of the
productspaceits βth co-ordinate,
πβ ( (xα)α ϵ J ) = xβ
It is called the projection mapping associated with the index β.
Definition :
Let Sβ denote the collection
Sβ = { ∏-1
β (Uβ ) / Uβ open in Xβ }
And let S denote the union of these collections,
S = Uβ ϵ J Sβ
The topology generated by the sub basis S is called the product topology.
In this topology ∏αϵ J Xα is called a product space.
Theorem
Comparison of the Box and Producttopologies
Statement :
The box of topology on ∏ Xα has an basis all sets of the form ∏ Uα where
Uα is open in Xα for each α. The producttopologyon ∏ Xα has a basis all sets
13. of the form ∏ Uα where Uα is open in Xα for each α and Uα equals Xα except
for finitely many values of α.
Proof:
Basis for producttopology on ∏ Xα .
The collection B consistof all finite intersection of elements of S.
If we intersect elements belonging to the same collection of Sβ then
∏-1
β ( U β ) Ո ∏-1
β (V β ) = ∏-1
β (U β Ո Vβ )
Thus the intersection of two elements of Sβ or finitely many such elements is
again an element of Sβ .
So let us intersect elements from different sets Sβ . Let β1 , β2 , …… βn be a
finite set of distinct indices from the index set J .
Let Uβ be an open set in Xβi , i = 1, 2, …. n . Then
∏-1
β 1 ( U β 1 ) Ո ∏-1
β 2 ( U β 2 ) Ո …… Ո ∏-1
β n ( U β n)
is the finite intersection of subbasis elements so it belongs to B .
Let β = ∏-1
β 1 ( Uβ 1 ) Ո ∏ -1
β 2 ( Uβ 2 ) Ո …… Ո ∏-1
β n ( Uβ n)
Let x = ( x α )α ϵ J ϵ β
⇔ ( x α )α ϵ J ϵ ∏-1
β 1 ( U β 1 ) Ո ∏-1
β 2 ( U β 2 ) Ո …… Ո ∏ -1
β n ( Uβ n)
⇔ ( x α )α ϵ J ϵ ∏ -1
β i ( Uβ i ) , i = 1, 2, …… n
⇔ ∏ β i (( x α ))α ϵ J ϵ Uβ i
⇔ Xβ i ϵ Uβ i
There is no intersection on αth co-ordinates of x if α is not one of the indices
β1 , β2 , …… βn
x ϵ B ⇔ ∏ Uα
Where U α is open in X α for all α and V α = X α if α ≠ β1 , β2 , …… βn .
Thus ∏ Xα has as basis all sets of the form ∏ U α where U α is open in X α
for each α and U α equals X α except for finitely many values of α .
Hence the theorem
14. Example :
i) For finite product ∏ X α
𝑛
α=1 the two topologies are precisely the
same .
ii) The boxtopology is in general finer than the producttopologyfor
any basis element of the form ∏ Uα where Uα is open in Xα is
contained in ∏ U α ,where U α is open in X α for each α and Uα
equals X α except for finitely many values of α .
Theorem :
Supposethe topology on each space X α is given by a basis ℬα . The
collection of all sets of the form ∏α = J Bα where Bα ϵ ℬα for each α will
serve as a basis for boxtopology on ∏α = J Xα .
The collection of all sets of the same form, where B α ϵ ℬα for finitely
indices α and Bα = Xα for all the remaining indices , will serve as a basis for
the producttopology ∏α= J X α
Proof:
BoxTopology
Let ( ( X α ) )α = J ϵ W and W be an open set in ∏α= J X α . Forbox
topology on ∏X α there exist a basis element ∏α = J U α where each U α open
un X α such that
(Xα ) ϵ ∏ Uα ⊂ W
Since ℬα generates Xα , for each X α ϵ U α there exist Bα ϵ ℬα such that
xα ϵ Bα ⊂ U α .
Hence (x α )α = J ϵ ∏ B α ⊂ ∏ U α ⊂ W
Hence by theorem “ Let ( X , 𝜏 ) be a topological space. Supposethat C
is a collection of open sets of X such that for each open set U of x such that
x ϵ C ⊂ U . Then C is a basis for the topological basis for the box topology on
∏α = J Xα “ .
Product Topology:
Take U = ∏ Uα
Let (Xα )α = J ϵ W and W be an open set in ∏α = J X α . For the product
topology on ∏ X α there exist a basis element ∏α = J Uα where each Uα is
open in Xα and Uα = X α except for finitely many α ,s.
15. (x α ) ϵ U ⊂ W
Since ℬα generates X α for each x α ϵ Uα there exist Bα ϵ ℬα such that
xα ϵ B α U α ( except for finitely many α’s note that Uα = Xα )
Hence (x α ) ϵ ∏ B α ⊂ ∏ Uα = U ⊂ W
Hence by above stated theorem ∏α = J Bα where Bα ϵ ℬα for each α is a
basis for the producttopology on ∏ X α .
Hence the proof.
Example :
Consider Euclidean n – space ℝn . A basis for ℝ consists of all open
intervals in ℝ , hence a basis for the topology of ℝn consists ofall products of
the form
( a 1 , b 1 ) x ( a 2 , b 2) x ………. x ( a n , b n )
Since ℝn is a finite product, the box and producttopologies agree whenever
we consider ℝn , we will assume that it is given this topology unless we
specifically state otherwise.
Theorem
Let Aα be a subspaceof Xα for each α ϵ J, then ∏ Aα is a subspaceof
∏ X α if bothproducts are give the box topology or if both products are given
the producttopology.
Proof:
Box Topology:
Consider ∏ Xα and ∏ Aα with box topologies . Let ∏ Uα , Uα is open
in Xα for all α be a general basis element of ∏ Xα. . That implies Uα ⊆ X α
for all with Uα is open in X α,
→ U α Ո Aα ⊆ Aα is open in Aα Since each A α is a subspace
→ ∏α = J ( Uα Ո A α ) ⊆ ∏Aα is a basis element for ∏Aα
But ∏ ( Uα Ո A α ) = ( ∏ Uα ) Ո ( ∏ Aα )
Therefore ( ∏ Uα ) Ո ( ∏ Aα ) is a basis of ∏ Aα with ∏ Uα is basis element
for ∏ Xα . So ∏ Aα is a subspaceof∏ Xα in box topology.
16. Product Topology:
Supposebothproducts are given producttopologies.
Claim :
∏ Aα is a subspaceof ∏ Xα . Let ∏ Uα be a general basis element of ∏ Xα
where Uα is open in Xα .
For finitely many α’s say β1 , β2 , …… βn and Uα = Xα for the remaining
α’s. Since each Aα is a subspaceofXα , Uβ i Ո A β i is open in A β i , i = 1, 2,
…. n and Xα Ո Aα is open in Aα for the remaining α’s.
Let Vα = Xα Ո Aα if α = βi , i = 1, 2, ……. n
Aα if α ≠ β i
Then ∏ Vα is a general basis element of ∏ Aα . Therefore ∏ Aα is a
subspaceof ∏ Xα in producttopology.
Hence the proof.
Theorem:
If each space Xα is Hausdorff spacethen ∏ Xα is Hausdorff space in
both the box and producttopologies.
Proof:
Claim : ∏α= J X α is Hausdorff
Let (xα )α = J ≠ (yα ) α = J in ∏α = J Xα . Then there exist atleast one β ϵ J such
that x β ≠ y β . Now X β is a Hausdorff and xβ ≠ y β in Xβ .
There exist two opensets Uβ and Vβ in Xβ such that xβ ϵ Uβ , yβ ϵ Vβ
and Uβ Ո Vβ = Φ
Now consider the projection
∏ β : ∏ X α → Xβ
∏-1
β ( Uβ ) is open in ∏ Xα and ( xα ) ϵ ∏-1
β ( Uβ )
Similarly,
∏-1
β ( Vβ ) is open in ∏ X α and ( y α ) ϵ ∏-1
β ( Vβ ) and
∏-1
β ( Uβ ) Ո ∏-1
β ( Vβ ) = ∏-1
β ( Uβ Ո Vβ )
= Φ
In either topology this result holds good. If ∏ X α is given box topology then
17. ∏-1
β ( Uβ ) = ∏ U α where U α is open in Xα foll all α .
If ∏ X α is given producttopology then ∏-1
β ( Uβ ) = ∏ Uα where U α is
open in Xα for finitely many α’s and Uα = X α for the remaining α’s.
Hence the proof.
Theorem :
Let { X α } be an indexed family of spaces. Let Aα ⊂ Xα for each α .
Then ∏ Xα is given either the producttopology or the box topology , then
∏Āα =∏ A α
Proof:
∏Āα = ∏Aα
x = ( xα ) ϵ ∏Āα
Claim :
x ϵ ∏Aα
Let U = ∏ Uα be a basis element for either topology that contain x.
x α ϵ U α for all α and so U α intersects Aα as xα ϵ Āα .
Let yα ϵ U α Ո Aα for all α . Then y = ( y α ) ϵ ∏ U α and ∏ A α .
Since U is arbitrary, every basis element about x intersects ∏ A α .Therefore
x ϵ ∏Aα → ∏Āα ⸦ ∏Aα
Conversely ,
Let x = ( xα ) ϵ ∏Āα in either topology.
To Prove :
xα ϵ Aα , for all α .
So that ( xα ) ϵ ∏Āα choosea particular index β .
To Prove :
x β ϵ A β
Let Vβ be an open set of X β containing x β .
∏-1
β ( Vβ ) is open in ∏ X α in either topology and X α ϵ ∏-1
β ( Vβ ) .
Therefore ∏-1
β ( Vβ ) Ո ∏ A α ≠ Φ
Let y = ( y α ) ϵ ∏-1
β ( Uβ ) Ո ∏ Aα for the index β , yβ ϵ Aα
18. ( yα ) ϵ ∏-1
β ( V ) → ( yα ) ϵ Vβ
ie , yα ϵ Vβ
ie , yα ϵ Vβ Ո Aβ
Therefore ( Xα ) ϵ ∏Āα
→ ∏ Aα = ∏Āα
Hence ∏Āα = ∏ Aα
Hence Proved
Theorem:
Let f : A → ∏ Xα be given by the equation f (a) = ( fα (a) )α ϵ J
where fα : A → Xα for each α . Let ∏ Xα have the product topology .
Then the function f is continuous if and only if each function fα is continuous .
Proof:
Let f : A → ∏ Xα is given by the equation f (a) = ( fα (a) )α ϵ J where
fα : A → Xα for each α .
Claim :
fα is continuous .
Let ∏β be the projection of the product onto it’s the βth factor. The function
∏β is continuous.Now, suppose that the function f: A → ∏ Xα is continuous.
The function f β equals the composite ∏β o f being the composite of two
continuous functions is continuous.
Conversely, suppose that each fα is continuous. To prove that f is continuous
, it is enough to prove that inverse image of every sub-basis element is open in A.
A typical sub basis element for the product topology on ∏ Xα is a set of the
form π-1
β ( Uβ ), where β is some index and Uβ is open in Xβ.
Now , f-1 ( π-1
β ( Uβ ) ) = ( πβ 0 f )-1 ( Uβ )
= f-1
β (Uβ )
Because fβ = πβ 0 f . Since fβ is continuous, this set is open in A.
→ f is continuous
Hence the proof.
Note :
The above theorem fails if ∏ Xα is given boxtopology.
19. Example :
Consider ℝw be the countably infinite productof ℝ with itself recall
that,
ℝw = ∏ n ϵ ℤ + Xn
Where Xn ϵ ℝn for each n. Let us define a function f : ℝ → ℝw by the
equation
f ( t ) = ( t , t, t, …. )
The nth co – ordinate function f is the function fn ( t ) = t .Each of the
coordinate functions fn : ℝ → ℝ is continuous .
Therefore the function f is continuous if ℝw is given by the product
topology . But f is not continuous if ℝw is given by the box topology.
Consider the example , the basis element
B = ( -1, 1 ) (-½ , ½ ) ( -⅓ , ⅓ ) ……….
for the box topology , we assert that f-1 ( B ) is not open in ℝ.
If f-1 ( B ) were open in ℝ , it would contain some interval (- δ , δ ) about
the point U . This means that f (- δ , δ ) ⸦ B , so that applying πn to both sides
of the induction ,
fn (- δ , δ ) = (- δ , δ ) ⸦ (- 1/n, 1/n )
for all n, a contradiction.
Metric Topology:Definition :
A metric on a set X is a function d : X x X → ℝ having the
following properties:
1. d ( x, y ) ≥ 0 for all x, y ϵ X ; equality holds iff x = y
2. d ( x, y ) = d ( y, x ) for all x, y ϵ X
3. ( Triangle inequality ) d ( x, y ) + d ( y,z ) ≥ d ( x, z ) for all x , y, z
ϵ X
Example: 1
Given a set , define d ( x, y ) = 1 if x ≠ y
d ( x, y ) = 0 if x = y
Its trivial to check that d is a metric .
20. The topology induced is the discrete topology . The basis element
B ( x , 1 ), for example, consistof the point x alone .
Example : 2
The standard metric on the real number ℝ is defined by
d ( x, y ) = │x – y │
d ( x, y ) ≥ 0 iff x=y
│x – y │ = │y- x │ and
d ( x, z ) = │x – z │ = │x – y + y - z│ ≤ │x – y │ + │y - z │
= d(x,y) + d(y,z)
Definition :
Given a metric d on X the number d ( x, y ) is often called the
distance between x and y in the metric d.
Given ϵ > 0 , consider the set , Bd ( x, ϵ ) = { y / d ( x, y ) < ϵ } of all points
y whose distance from x is less than ϵ . It is called the ϵ - ball centered at x .
Note :
In ℝ the topology induced by the metric d ( x, y ) = │x – y│ is the
same as the order topology. Each basis element ( a, b ) for the order topology
is a basis element for the metric topology. Indeed ( a, b ) = B( x , ϵ )
where x = a+b/2 and ϵ = b-a/2 and conversely , each ϵ- ball B( x , ϵ) equals an
open interval ( x-ϵ, x+ϵ ) .
Definition :
Metric Topology:
If d is a metric on the set X, then the collection of all ϵ - ball
Bd ( x, ϵ ) for x ϵ X and ϵ > 0 is a basis for a topology on X , called
the metric topology induced by d .
Result 1 :
If y is a point of the basis element B ( x, ϵ ) , then there is a basis
element B ( y, δ ) centered at y that is contained in B ( x, ϵ ) .
Proof:
21. Define δ to be the positive number ϵ - d(x, y). Then
B ( y, δ ) < B ( x, ϵ ) for if z ϵ B ( y, δ ) then d ( x, z ) < ϵ - d(x, y) from
which we conclude that
d ( x, z ) ≤ d ( x, y ) + d ( y, z ) < ϵ
Hence the result .
Result 2 :
B = { Bd ( x, ϵ ) / x ϵ X and ϵ > 0 } is a basis.
Proof:
First condition for a basis :
x ϵ B ( x, ϵ ) for any ϵ > 0.
Second condition for a basis :
Let B1 and B2 be two basis elements . Let y ϵ B1 Ո B2 . We can choosethe
number δ1 > δ2 so that B(y, δ1 ) ≤ B1 and B(y, δ2 ) ≤ B2 .
Letting δ be the smaller of δ1 and δ2 . We can conclude that
B(y, δ ) ≤ B1 Ո B2 .
Result 3 :
A set U is open in the metric topology induced by d iff for each y ϵ U ,
there is a δ > 0 such that Bd (y, δ ) ⊆ U .
Proof:
Clearly this condition implies that U is open .
Conversely , If U is open it contains a basis element B = Bd ( x, ϵ ) containing
y, and B in turn contains a basis element Bd (y, δ ) centered at y.
Hence the result.
Definition :
If X is a topological spaces . X is said to be metrizable if there exist a metric
d on the set X that induces the topology of X . A metric space is a metrizable
space X together with a specific metric d that gives the topology of X .
22. Definition :
Let X be a metric space with metric d . A subset A of X is said to be
bounded if there is some number M such that d (a1 , a2) ≤ M for every pair a1 , a2
of points of A .
Definition :
If A is bounded and non – empty the diameter of A is defined to be the
number .
diam A = sup {d (a1, a2) / a1, a2 ϵ A }
Theorem :
Let X be a metric spacewith metric d . Define đ : X x X → ℝ by the
equation
đ ( x, y ) = min { d ( x, y ), 1 }
Then đ is a metric that induces the same topology as d . The metric đ is called
the standard bounded metric correspondingto d .
Proof:
d is a metric .
đ ( x, y ) = min { d ( x, y ), 1 } ≥ 0
đ ( x, y ) = 0 ↔ min { d ( x, y ), 1 } = 0
↔ d ( x, y ) = 0
↔ x = y ( d is a metric )
đ ( x, y ) = min { d ( x, y ), 1 }
= min { d ( x, y ), 1 }
= đ ( y, x )
Claim :
đ ( x, z ) ≤ đ ( x, y ) + đ ( y, z )
Supposeđ ( x, y ) = d ( x, y ) , đ ( y, z ) = d ( y, z )
And d( x, z ) ≤ d ( x, y ) + d ( y, z ) = đ ( x, y ) + đ ( y, z )
Also đ ( x, z ) ≤ d ( x, z ) ( by defn )
đ ( x, z ) ≤ đ ( x, y ) + đ ( y, z )
23. Supposed ( x, y ) ≥ 1 or d ( y, z ) ≥ 1 then
R.H.S of our claim is atleast 1 and L.H.S of our claim is atmost 1
The equality holds.
Hence đ is metric space.
Hence the theorem
Definition :
Given x = ( x1, x2, ….. xn ) in ℝn , we define the norm of x
by the equation
║x ║ = ( x1
2 + x2
2 + ….. + xn
2 ) ½
The euclidean metric d on ℝn is given by the equation
d (x, y) = ║x- y║= [ ( x1 - y1)2 + ( x2 – y2)2 + ….+(xn – yn)2 ]1/2
= [ ∑n
i=1 ( xi – yi)2 ]1/2
Where x = ( x1, x2, ….. xn ) , y = ( y1, y2, ….. yn ) .
Definition :
The square metric ρ on ℝn is given by the equation
ρ( x, y ) = max { | x1 - y1| , | x2 – y2| , …… , | xn – yn| }
Relationbetweeneuclidean metric d and sequence metric ρ in ℝn
is
ρ( x, y ) ≤ d( x, y ) ≤ √n (ρ ( x, y ))
Theorem:
Let d and d′ be two metrices on the set X . Let τ and τ′ be the
topologies they induce respectively. Then τ′ is finer than τ iff for each x in X
and each ε > 0 there exist a δ > 0 such that
Bd′ ( x, δ ) ⊂ Bd ( x, ε )
Proof:
Supposeτ ⊂ τ′
Then by lemma
“ Let ℬ and ℬ′ be basis for the topologies τ and τ′ respectively on X .
Then the following are equivalent
(i) τ ⊂ τ′
24. (ii) For each x ϵ X and each basis element B ϵ ℬ containing x , there is a
basis element B′ ϵ ℬ ′ such that x ϵ B′ ⸦ ℬ ′′.”
Given the basis element
Bd (x , ε) for τ, by the lemma, there exist a basis element B′ for the topology
τ′ such that
x ϵ B′ ⊂ Bd (x ,ε )
within B′ we can find a ball B′( x , δ)such that
x ϵ Bd′ ( x, δ ) ⊂ B′ ⊂ Bd ( x, ε )
Conversely ,
Supposethat ϵ -δ condition holds. given a basis element B for τ
containing x , we can find within B a ball Bd ( x, ε ) contained at x .
ie , x ϵ Bd ( x, ε ) ⊂ B
By hypothesis , x ϵ Bd ( x, δ ) ⊂ Bd ( x, ε )
By lemma , τ′ is finer than τ .
Hence the theorem
Theorem :
The topologies on ℝn induced by the euclidean metric d and the
square metric δ are the same as the producttopology on ℝn .
Proof:
Let X = ( x1, x2, ….. xn ) and y = ( y1, y2, ….. yn ) be two points of ℝn
.
Let τd be the topology induced by the metric d and τρ be the topology ,
induced by the metric ρ.
Claim : τρ = τd
To prove :
τρ ⊆ τd and τd ⊆τρ
ie , To prove : i) Bd ( x, ε ) ⊆ Bρ ( x, ε )
ii)Bρ ( x, ε /√n ) ⊂ Bd ( x, ε )
i) Let y ϵ Bd ( x, ε )
→ d( x, y ) < ϵ
25. → ρ( x, y ) < ϵ ( ρ( x, y ) ≤ d( x, y ) < ϵ )
→ y ϵ Bρ ( x, ε )
→ Bd ( x, ε ) ⊆ Bρ( x, ε )
Therefore τρ ⊆ τd ( by theorem )
ii) Let y ϵ Bρ ( x, ε /√n )
→ ρ( x, y ) < ϵ /√n
→ √n ρ( x, y ) < ϵ
→ d( x, y ) < ϵ
→ y ϵ Bd ( x, ε )
→ Bρ ( x, ε /√n ) ⊆Bd ( x, ε )
→ τd ⊆ τρ
We get , τd = τρ
Claim :
τ = τρ
To prove that the producttopology is same as the topology induced by the
square metric ρ .
First let prove τ ⊆ τρ , where τ is the producttopology on ℝn . Let
B = (a1, b1) x (a2, b2) x …….. x (an, bn) be a basis element of τ with x ϵ X
where X = ( x1, x2, ….. xn ).
Now for each i there is an εi such that
( xi - εi , xi + εi ) ⊆ (ai , bi)
Thus Bρ ( x, ε ) ⊆ B for y ϵ Bρ ( x, ε )
→ ρ( x, ϵ ) < ϵ
→ max { |x1 – y1| , |x1 – y1|, …… |xn – yn| } < ϵ
→ |xi – yi| < ϵ < ϵi , for all i = 1, 2, ….. n
→ yi ϵ ϵ ( xi - εi , xi + εi ) for all i
→ yi ϵ ϵ (ai , bi) for all i
→ y ϵ B
→ τ ⊆ τρ ……… (1)
Conversely ,
To Prove :
τρ ⊆ τ
26. Let Bρ ( x, ε ) be a basis element for ρ topology given the element
y ϵ Bρ ( x, ε )
We need to find a basis element B for producttopology such that
y ϵ B ⊆ Bρ ( x, ε )
Now Bρ ( x, ε ) = ( (x1 - ε, x1 + ε ) (x2 - ε , x2 + ε ) ……. (xn - ε , xn + ε ) )
Which is itself a basis element of the producttopology .
y ϵ B = Bρ ( x, ε )
Hence τρ ⊆ τ …….. (2)
From (1) and (2)
τρ = τ
Definition :
Given a indexed set J and given points x = ( xα )α ε J and
y = ( yα )α ε J of RJ , a metric ρ on RJ defined by the equation
ρ ( x, y ) = sup { đ ( xα , yα ) } α ε J
where đ ( x, y ) = min { |x – y| , 1 } the standard bounded metric on ℝ.
ρ is a metric on RJ called the uniform metric on RJ . The topology it
induces is called the uniform topology.
Theorem:
The uniform topology on RJ is finer than the producttopology and coarser
than the box topology.
Proof:
( i ) Let τρ be the producttopology on RJ . τB be box topology on RJ and τρ
be the uniform topology on RJ . The theorem states that
τρ ⊆ τρ ⊆ τB
First let us prove :
τρ ⊆ τρ
Let x = ( xα )α ε J and U = Π Uα be a basis element of τρ with ( xα )α ε J ϵ U
Let α1 , α2 , α3 , ….. αn be the induces for which Uα ≠ R then for each i , we
can choosenan εi > 0 such that
27. B d (xi , εi ) ⊆ Ui for all i = α1 , α2 , α3 , ….. αn
Where Ui is open in R .
Let ε = min { ε1 , ε2 , ….. εn }
Then Bρ (x ,ε ) ⊆ U for Z ϵ Bρ (x ,ε )
→ ρ ( x , z ) < ε
→ sup { đ ( xα , zα ) }α ε J < ε
→ d ( xα , zα ) < ε < εi for all α
→ zα ϵ Bd ( xα , ε )
→ Z ϵ U
→ Z ϵ Π Uα
Thus τρ ⊆ τρ
( ii ) To Prove :
τρ ⊆ τρ
Let Bp ( x, ε ) be a basis element of τρ . Then the neighbourhood
U = Π ( xα - ε / 2 , xα + ε / 2 ) is x ϵ U ⊆ Bp ( x, ε ) for y ϵ U
→ Uα ϵ ( xα - ε / 2 , xα + ε / 2 ) for all α
→ d ( xα , yα ) < ε / 2 for all α
→ sup { d ( xα , yα ) } α ε J < ε / 2
→ p (x ,y) < ε / 2 < ε
→ y ϵ Bp ( x, ε )
The topologies τρ , τρ and τB in RJ are different if J is infinite.
Theorem:
Let d ( a,b) = min { |a – b| , 1 } be the standard bounded metric on R . If x
and y are two points of Rw , Define
D ( x,y ) = sup { d ( xi , yi ) / i }
Then D is a metric that induces the producttopology on Rw
Proof:
D ( x,y ) = sup { d ( xi , yi ) / i } is a metric.
28. Since each 𝑑̅(𝑥𝑖,𝑦𝑖) ≥ 0,𝐷(𝑥, 𝑦) ≥ 0
𝐷(𝑥, 𝑦) = 0 ⇔ 𝑑
̅(𝑥𝑖,𝑦𝑖) = 0 ∀ 𝑖
⇔ 𝑥𝑖 = 𝑦𝑖 ∀ 𝑖
⇔ 𝑥 = 𝑦
𝐷(𝑥, 𝑦) = 𝑠𝑢𝑝 {
𝑑̅(𝑥𝑖, 𝑦𝑖)
𝑖
} = sup{
𝑑̅(𝑦𝑖, 𝑥𝑖)
𝑖
}
= 𝐷(𝑥,𝑦)
Since 𝑑 is metric
𝑑̅(𝑥𝑖,𝑧𝑖) ≤ 𝑑̅(𝑥𝑖,𝑦𝑖) + 𝑑̅(𝑦𝑖, 𝑧𝑖)
≤
𝑑
̅(𝑥𝑖,𝑦𝑖)
𝑖
≤
𝑑
̅(𝑥𝑖,𝑦𝑖)
𝑖
≤
𝑑
̅(𝑦𝑖 ,𝑧𝑖)
𝑖
≤ 𝑠𝑢𝑝{
𝑑̅(𝑥𝑖,𝑦𝑖)
𝑖
} + 𝑠𝑢𝑝{
𝑑̅(𝑦𝑖, 𝑧𝑖)
𝑖
}
𝑑̅(𝑥𝑖, 𝑧𝑖) ≤ 𝐷(𝑥, 𝑦) + 𝐷(𝑦,𝑧)
𝑠𝑢𝑝{
𝑑̅(𝑥𝑖,𝑧𝑖)
𝑖
} ≤ 𝐷(𝑥, 𝑦) + 𝐷(𝑦,𝑧)
∴ 𝐷 is a metric on ℝ.
Claim: 𝜏𝐷 = 𝜏𝑝 (we have topology induced by 𝐷 be 𝜏𝐷
To prove: 𝜏𝐷 ⊆ 𝜏𝑝
Let U be open in metric topology 𝜏𝐷 and 𝑥 ∈ 𝑈
To prove that ∃ an open set V in 𝜏𝑝 ∋ 𝑥 ∈ 𝑉 ⊆ 𝑈.
Since 𝑥 ∈ 𝑈, ∃ 𝑎 𝜀 > 0, 𝐵𝐷 (𝑥, 𝜀) ⊂ 𝑈.
ChooseN large enough 1
𝑁
⁄ < 𝜀.
Let 𝑉 = (𝑥1 − 𝜀, 𝑥1 + 𝜀) × (𝑥2 − 𝜀, 𝑥2 + 𝜀) × ………. .× (𝑥𝑁 − 𝜀, 𝑥𝑁 + 𝜀)
We assert that 𝑉 ⊂ 𝐵𝐷 (𝑥, 𝜀) for if 𝑦 ∈ 𝑅𝑤
then
𝑑
̅(𝑥𝑖,𝑦𝑖)
𝑖
< 1
𝑁
⁄ ∀ 𝑖 > 𝑁
∴ 𝐷(𝑥, 𝑦) = 𝑚𝑎𝑥 {
𝑑̅(𝑥1,𝑦1)
1
,
𝑑̅(𝑥2,𝑦2)
2
, ……… ……. .
𝑑̅(𝑥𝑁,𝑦𝑁 )
𝑁
}
If 𝑦 ∈ 𝑉 then 𝐷(𝑥, 𝑦) < 𝜀 ⇒ 𝑦 ∈ 𝐵𝐷 (𝑥,𝜀)
29. ∴ 𝑥 ∈ 𝑉 ⊂ 𝐵𝐷 (𝑥, 𝜀) ⊂ 𝑉
𝜏𝐷 ⊆ 𝜏𝑝
Consider the basis element 𝑈 = ∏ 𝑈𝑖
𝑗∈𝑍+
for the producttopology, where 𝑈𝑖 is
open in R for 𝑖 = (𝛼1, 𝛼2, ……. . 𝛼𝑛) and 𝑈𝑖 = 𝑅 for all other indices i.
Let 𝑥 ∈ 𝑈. Let us find an set V of the metric topology such that
x ∈ V ⊂ U. Choosean interval (𝑥𝑖 − 𝜀𝑖 ,𝑥𝑖 + 𝜀𝑖) in R centered about xi and
lying in Ui, for i = α1, . . . , αn; chooseeach 𝜀𝑖 ≤ 1.
Then define
𝜀= min{𝜀𝑖 /i | i = α1, . . . , αn}.
𝑥 ∈ 𝐵𝐷 (𝑥, 𝜀)⊂ U.
For y ∈ 𝐵𝐷 (𝑥, 𝜀),
⇒ 𝐷(𝑥, 𝑦) < 𝜀 ⇒
𝑑
̅(𝑥𝑖,𝑦𝑖)
𝑖
≤ 𝐷(𝑥, 𝑦) < 𝜀 for all i ,
⇒ 𝑑̅(𝑥𝑖,𝑦𝑖) ≤ 𝐷(𝑥, 𝑦) < 𝜀(𝑖) < 𝜀𝑖 for i = α1, . . . , αn
⇒ min{|𝑥𝑖 − 𝑦𝑖|,1} < 𝜀1 < 1
⇒ 𝑦𝑖 ∈ (𝑥𝑖 − 𝜀𝑖 ,𝑥𝑖 + 𝜀𝑖) ⊂ 𝑈𝑖 ∀ i = α1, . . . , αn
obviously, 𝑦𝑖 ∈ 𝑅 for other indices i,
∴ 𝑦 ∈ 𝑈
Hence, 𝑥 ∈ 𝐵𝐷 (𝑥, 𝜀)⊂ U (basis element of 𝜏𝑃)
∴ 𝜏𝑃 ⊆ 𝜏𝐷
Hence 𝜏𝑃 = 𝜏𝐷
Hence the theorem.
Theorem:
Let 𝑓 ∶ 𝑋 → 𝑌. Let 𝑋 and 𝑌 bemetrizable with metrics 𝑑𝑋 and 𝑑𝑌, respectively.
Then continuity of 𝑓 is equivalent to the requirement that given 𝑥 ∈ 𝑋 and given
ε > 0, there exists 𝛿 > 0 such that
𝑑𝑋(x, y)< 𝛿 ⇒ 𝑑𝑌 ( f (x), f (y)) < 𝜀.
30. Proof.
Supposethat f is continuous.
Given x and 𝜀, consider the set
𝑓−1
(𝐵( 𝑓 (𝑥),𝜀)),
which is open in X and contains the point x.
∴ ∃ 𝛿 > 0 ∋ 𝐵(𝑥,𝛿) ⊂ 𝑓−1
(𝐵( 𝑓 (𝑥),𝜀) )
Then 𝑦 ∈ 𝐵(𝑥, 𝛿) ⇒ 𝑦 ∈ 𝑓−1
𝐵( 𝑓 (𝑥), 𝜀))
i.e 𝑦 ∈ 𝐵(𝑥, 𝛿) ⇒ 𝑓(𝑦) ∈ 𝐵( 𝑓 (𝑥),𝜀)
i.𝑒 𝑑𝑋(𝑥, 𝑦)< 𝛿 ⇒ 𝑑𝑌 ( f (x), f (y)) < 𝜀.
Thus 𝜀 − 𝛿 conditions holds
Conversely,
Suppose𝜀 − 𝛿 conditions holds
Claim:
𝑓 ∶ 𝑋 → 𝑌 is continuous
Let V be open in Y; we
show that f −1(V) is open in X. Let 𝑥 ∈ 𝑓 −1
(𝑉). Then 𝑓 (𝑥) ∈ 𝑉,
As V is open, there exists 𝜀 > 0 such that 𝐵( 𝑓 (𝑥),𝜀) ⊆ 𝑉
By the 𝜀 -δ condition, there is a δ-ball B(x, δ) centered at x such that f (B(x, δ))
⊂ B( f (x), 𝜀).
⇒ Thus 𝑥 ∈ B(x, δ) contained in f −1(V), so that f −1(V) is open in X.
Hence continuity and 𝜀 -δ conditions are equivalent.
Hence the theorem.
The Sequence lemma :
Let X be a topological space;let A ⊂ X. If there is a sequence of points of A
converging to x, then x ∈ 𝐴̅; the converse holds if X is metrizable.
Proof:
Supposethat xn → x, where xn ∈ A then for every neighborhood of x
contains a point of A ⇒ 𝑥 ∈ 𝐴̅ ̄
by Theorem “Let A be a subset of the topological spaceX.
31. Then 𝑥 ∈ 𝐴̅ iff every open set V containing x intersects A.
Supposing the topology of X is given by a basis then 𝑥 ∈ 𝐴̅ ̄iff every basis
element B containing x intersects A”
Conversely, supposethat X is metrizable and 𝑥 ∈ 𝐴̅. Let d be a metric for the
topology of X. For each positive integer n, take the neighborhood Bd (x, 1/n) of
radius 1/n of x, and choosexn ∃ xn ∈ 𝐵𝑑 (𝑥,
1
𝑛
)
Then (xn ) is a sequence of A.
Claim: xn converges to x
Any openset U containing x is such that 𝑥 ∈ 𝐵(𝑥, 𝜀) ⊂ 𝑈
if we chooseN so that 1/N < 𝜀 , then
1
𝑁+3
<
1
𝑁+2
<
1
𝑁+1
<
1
𝑁
< 𝜀 and so that U
contains xi for all i ≥ N.
Hence the proof.
Theorem:
Let f : X → Y. If the function f is continuous, then for every convergent
sequence xn → x in X, the sequence f (xn) converges to f (x). The converse
holds if X is metrizable.
Proof:
Assume that f is continuous. Let xn → x.
Claim: f (xn) → f (x).
Let V be a neighborhood of f (x). Then f −1(V) is a neighborhood of x, and so
there is an N suchthat xn ∈ f −1(V) for n ≥ N.
Then f (xn) ∈ V for n ≥ N.
f (xn) → f (x)
conversely,
Suppose that the convergent sequence condition is satisfied.
Let X is metrizable and A ⊂ X;
To prove : f is continuous
32. We prove, f (𝐴̅) ⊆ 𝑓 (𝐴)
̅̅̅̅̅̅̅. Let x ∈ 𝐴̅, then there is a
sequence xn in x such that xn → x (by the sequence lemma).
By hypothesis, f (𝑥𝑛) converges to f (x). As 𝑓(𝑥𝑛)) is in (𝐴) ; f (xn) ∈ 𝑓 (𝐴)
̅̅̅̅̅̅̅
Hence f (𝐴̅) ⊆ 𝑓 (𝐴)
̅̅̅̅̅̅̅
i.e, f is continuous
Hence the theorem.
Theorem:
The addition, subtraction and multiplication operations are continuous
function from ℝ × ℝ into ℝ; and the quotient operation is a continuous function
from ℝ × (ℝ − {0}) into ℝ.
Proof:
Let the addition operation be defined by
𝑓
1: ℝ × ℝ → ℝ , 𝑓
1(𝑥, 𝑦) = 𝑥 + 𝑦
The multiplication operation be defined by
𝑓
2 : ℝ × ℝ → ℝ , 𝑓2(𝑥, 𝑦) = 𝑥𝑦
Let us use the metric:
𝑑(𝑎, 𝑏) = |𝑎 − 𝑏| on ℝ and let the metric on ℝ2
given by,
𝜚((𝑥, 𝑦),(𝑥0,𝑦0)) = max{|𝑥 − 𝑥0|,|𝑦 − 𝑦0 |}
Let (𝑥0,𝑦0 ) ∈ ℝ × ℝ and 𝜀 > 0 be given if 𝜚((𝑥,𝑦), (𝑥0,𝑦0)) < 𝛿. Then,
max{|𝑥 − 𝑥0|,|𝑦 − 𝑦0| < 𝛿
i.e., |𝑥 − 𝑥0| < 𝛿 and |𝑦 − 𝑦0 | < 𝛿 .
To prove: 𝑓
1 is continuous
Choose𝛿 <
𝜀
2
𝑑(𝑓
1(𝑥, 𝑦),𝑓
1 (𝑥0,𝑦0)) = |𝑓
1 (𝑥, 𝑦) − 𝑓
1(𝑥0,𝑦0)|
= |(𝑥 + 𝑦) − (𝑥0 + 𝑦0)|
= |𝑥 − 𝑥0 + 𝑦 − 𝑦0|
= |𝑥 − 𝑥0| + |𝑦 − 𝑦0 |
34. Theorem:
If Χ is a topological spaceand if 𝑓 ∘ 𝑔: Χ → ℝ are continuous functions,
then
𝑓 + 𝑔, 𝑓 − 𝑔 and 𝑓 ⋅ 𝑔 are continuous. If 𝑔(𝑥) ≠ 0 ∀𝑥 then
𝑓
𝑔
is continuous.
Proof:
The map ℎ: Χ → ℝ × ℝ defined by ℎ(𝑥) = (𝑓(𝑥),𝑔(𝑥)) is continuous by
the theorem, “maps into products”, ‘Let 𝑓: A → 𝑋 × 𝑌 be given by the equation
𝑓(𝑎) = ( 𝑓
1(𝑎), 𝑓
2 (𝑎))
Then 𝑓 is continuous iff if the function 𝑓
1: 𝐴 → 𝑋 and 𝑓
2 : 𝐴 → 𝑌 are continuous.
The maps 𝑓
1 and 𝑓2 are called the co-ordinate functions of 𝑓’.
The function 𝑓 + 𝑔 = 𝑓
1 ∘ ℎ where ℎ:𝑋 → ℝ × ℝ , 𝑓
1 : ℝ × ℝ → ℝ
𝑥 → (𝑓(𝑥), 𝑔(𝑥)) , (𝑓(𝑥),𝑔(𝑥) → 𝑓(𝑥) + 𝑔(𝑥)
(𝑓 + 𝑔)𝑥 = 𝑓(𝑥) + 𝑔(𝑥) , ∀𝑥 ∈ 𝑋
is continuous.
Since 𝑓
1 is continuous (by previous lemma), ℎ is continuous and
composition of continuous function is continuous.
The function 𝑓𝑔 = 𝑓 ∘ ℎ defined by
𝑓𝑔(𝑥) = (𝑓2 ∘ ℎ)(𝑥) = 𝑓
2 (ℎ(𝑥))
= 𝑓
2 (𝑓(𝑥),𝑔(𝑥))
= 𝑓(𝑥)𝑔(𝑥)
is continuous.
Since 𝑓
1 is continuous, composition of continuous function is continuous.
Similarly, functions 𝑓 − 𝑔 and
𝑓
𝑔
(𝑔 ≠ 0) are continuous.
Definition:
Let 𝑓
𝑛: 𝑋 → 𝑌 be a sequence of functions from the set 𝑋 to the metric
space𝑌.
We say that the sequence {𝑓
𝑛} converges uniformly to the function 𝑓: 𝑋 → ℝ if
given 𝜀 > 0, there exists an integer ℕ suchthat
35. 𝑑(𝑓
𝑛(𝑥),𝑓(𝑥)) < 𝜀 , ∀𝑛 > ℕ and for all 𝑥 in Χ.
Uniform limit theorem
Let 𝑓
𝑛: 𝑋 → 𝑌 be a sequence of continuous functions from the topological
spaceΧ into the metric space 𝑌. If {𝑓
𝑛 } converges uniformly to 𝑓, then 𝑓 is
continuous.
Proof:
Let 𝑉 be openin 𝑌.
Let 𝑥0 be a point of 𝑓−1
(𝑉). We wish to find a neighborhood 𝑈 of 𝑥0
such that 𝑓(𝑈) ⊂ 𝑉.
Let 𝑦0 = 𝑓(𝑥0) first choose𝜀 so that the 𝜀-ball Β(𝑦0, 𝐸) is contained in
𝑉.
Then using the uniform convergence
Chooseℕ so that for all 𝑛 ≥ ℕ and all 𝑥 ∈ Χ
𝑑(𝑓
𝑛(𝑥),𝑓(𝑥)) <
𝜀
3
--- (1)
Finally, using continuity of 𝑓
𝑁 , choosea neighborhood 𝑈 of 𝑥0 such that
𝑓
𝑁 (𝑈) ⊆ Β(𝑓
𝑁(𝑥0),
𝜀
3
) --- (2)
Claim:
𝑓(𝑉) ⊂ 𝐵(𝑦0, 𝜀) ⊂ 𝑉
If 𝑥 ∈ 𝑈 then
𝑑(𝑓(𝑥),𝑓
𝑁 (𝑥)) <
𝜀
3
--- (3) (by the choice of 𝑁)
by (1)
𝑑(𝑓
𝑁 (𝑥),𝑓𝑁 (𝑥0)) <
𝜀
3
--- (4) (by the choice of 𝑉) by (2)
𝑑(𝑓
𝑁 (𝑥0),𝑓(𝑥0)) <
𝜀
3
--- (5) (by the choice of 𝑁)
by (1)
Adding and using the triangle inequality at 𝑥0 we see that
𝑑(𝑓(𝑥),𝑓(𝑥0)) <
𝜀
3
as desired.