The document presents a comprehensive overview of compactness in topology, outlining its significance and definitions, including the concept of compact spaces and subsequences in Euclidean spaces. It traces the historical development of topology and discusses key theorems, such as the Bolzano-Weierstrass theorem and Heine-Borel theorem, along with their implications in metric spaces. The document also explores related concepts like Hausdorff spaces, sequential compactness, and local compactness, emphasizing their equivalences and properties.

1. SHRI SHANKARACHARYA MAHAVIDYALAYA
JUNWANI,BHILAI NAGAR (C.G)
A PRESENTATION ON COMPACTNESS IN
TOPOLOGY M.S.II SEM. MATHS
ALKA DEVI
Department of Mathematics

2. Compactness
INTRODUCTION :
Compact generalized the idea of closed and bounded .
sequences in the set have limits inside the set . And no
sequences run off to infinity . consequently every sequence
has at least a convergent subsequence
The definition of compact is that every open cover has a
finite sub cover if you can find a collection of open sets whose
union contains your set, then just a finite number of those
open sets will cover your set.
Compactness is one of one of the most important features of
any topological spaces and plays the most significant role in
all of topology . Though in the general topological spaces
compactness in some what intangible and more in the
Euclidean spaces

3. History of compactness :
Mathematicians associate the emergence of topology as a
distinct field of mathematics with 1845 publication of
ANALYSIS SITUS by the Frenchman henri pointcare
alough many topological ideas has found their way into
mathematics during the previous century and a half. The
Latin phrase analysis situs may be translated as analysis of
position and is similar to the phrase geometria situs
meaning geometry of position used in 1735 by Swiss
mathematician Leonard Euler to describe his solution to
the Konigsberg bridge problem Euler's work on this
problem also cited as the beginning of graph theory the
study of network of vertices connected by edges which
shares many ideas with topology.

4. Compact space
In mathematics, specifically general topology and metric
topology, a compact space is an abstract mathematical space
whose topology has compactness property, which has many
important implication not valid in general spaces. Compactness
is not easy to describe precisely In an intuitive manner ; in some
sense it says that the topology allows the space to be considered
as” small”(compactness is a kind of topological counterpart to
finiteness of sets), even through as a set it may be quite larger
.then compactness means that whenever one choose infinitely
many sample point from the space, some of the samples must
eventually get arbitrarily close to at least one point of the space
More generally the Bolzano –weierstrass theorem characterizes
compactness of subspaces of the real numbers , or more
generally of finite dimensional Euclidean spaces, as those
subsets that are both closed and bounded.

5. Definition of compact space
Formally, a topological space x is called compact if each of its
open covers has a finite sub cover. Otherwise it is called non-
compact. Explicitly , this means that for every arbitrary
collection
{Uα}α ϵ A
of open subsets of x such that
X =U Uα
α ϵ A
Some branches of mathematics such as algebraic geometry,
typically influenced by the French school of Bourbaki, use the
term compact as a topological space that are both Hausdorff
and quasi compact .A single compact set is sometimes
referred to as a compactum following the Latin second
declension(neuter) the corresponding plural form is compact.

6. Compact subspace :
A subspace of topological space which is compact as a
topological space in is own rights is said to be compact
subspace.
Definition of Hausdorff space :
Points x and y in a topological space X can be separated by
neighborhood if there exists a neighborhood U of x and a
neighborhood V of y such that U and V are disjoint
(U∩ V=Ф) .X is a Hausdorff space if any two distinct points of
X can be separated by neighborhoods. This condition is the
third separation axiom which is why Hausdorff spaces are
also called T2 spaces. The name separated spaces is also
used.
.
x
.
y
U V

7. Equivalences of Hausdorff space :
For a topological space X the following are equivalent:
•X is a Hausdorff space.
• limits of nets in X are unique.
• limits of filters on X are unique
• Any singleton set {x} C X is equal to the intersection of all
closed neighborhoods of x .
The diagonal Δ ={ (x,x)| x ϵ X} is closed as a subset of product
spaces X x X
Properties of Hausdorff spaces :
Let f:X ―›Y be a quotient map with X a compact Hausdorff
space .then
The following are equivalent
. Y is Hausdorff
. F is closed map
. ker(f) is closed

8. Theorem 1: Closed subsets of compact sets are compact .
or
Every closed subspaces of compact space is compact
or
Let F be closed subset of compact space X then F is also
compact.
Proof: suppose Y be a compact subset of a topological space
X and let F be a subset of Y closed relative to X then we have
to show F is compact
Let C* ={G_ :ϵ}
be open cover of F then collection
D= {G_ :ϵ} F’
Forms the open cover of Y. since Y is compact than it has
finite sub-cover D of D which covers Y and hence F D and still
retain an open cover of F hence F is compact

9. Compactness and bases :
compactness is property that every open cover has a finite sub-
cover.
Definition (base ):
Let (X,T) be a topological space. A base for this space is a
collection B of open set in X can be expressed as the union of
sets in the base .the elements of b are referred to as basic open
sets.
Definition (sub-base):
Let (X ,T) be a topological space. A sub-base for this space is a
collection B subsets of X such that F is the weakest topology
that makes B open .Elements of b are referred to as sub basic
open sets.

10. Theorem 2 ( Alexander sub-base theorem) :
“ let(X,T) be a topological space with a sub-base B then the
following are equivalent:
Every open cover has a finite sub cover.
Every sub-basic open cover has a finite sub cover”.
Proof: call an open cover bad if it had no finite sub-cover,
and good otherwise . it suffices to show that if every sub-
basic open cover is good, then every basic open cover is
good also.
Suppose for contradiction that every sub-basic open
cover was good ,but at least one basic open cover was bad
.if we order the bad basic open cover by set inclusion,
observe that every chain of bad basic open cover, namely
the union of all the cover in the chain. Thus by Zorn’s
lemma, there exist a maximal bad basic open cover C
={Uα}α ϵ A

11. Thus this cover has no finite sub cover , but if one adds
new basic open set to this cover, then there must be no
finite sub cover
Pick a basic open set Uα
in this cover C . Then we can write Uα=B1 ∩..∩ Bk for some
sub-basic open setsB1 ∩..∩ Bk. We clam that at least one of
the B1 ∩..∩ Bk also lie in the cover C. To see this suppose
for contradiction that none of the B1 ∩..∩ Bk was in C. Then
adding any of the Bi to C enlarges the basic open cover and
thus creates finite sub cover ; thus Bi together with finitely
many sets from C cover X or equivalently that one can
cover XBi. thus one can also cover X Uα =U (XBi) and thus
X itself can be covered by finitely many says from C a
contradiction
From the above discussion and axiom of choice ,we see.
that for each basic set in there exist a sub-basic set
containing that also lies in c.(two basic sets could lead to
the same sub-basic set but this will, mot concern

12. us.)since the cover the do also. By hypothesis, a finite
number of can cover and also so is good, which gives the
desired a contradiction..

13. -
Bolzano weierstrass Theorem 3:
In real analysis th Bolzano weierstrass theorem is a
fundamental result about convergence in a finite dimensional
Euclidean space R
.the theorem states that each bounded sequence in R has a
convergent subsequence. An equivalent formulation is that a
subset of R is sequentially compact iff it is closed and
bounded.
LEMMA : every sequence {xn} in R has a monotone
subsequence.
Proof : let us call a positive integer n a peak of the sequence if
m>n implies xn>xm
suppose first that the sequence has infinitely
many peaks n1<n2<……nj<….then subsequence {xnj}
corresponding to peaks is monotonically decreasing and we
are done
so suppose now that there are only a finitely many peaks let

14. N be the last peak and n1=N+1.then n1 is not a peak since
n1>N which implies the existence of an n2>n1 with xn2>
xn1 .again n2>N is not a peak hence there is n3 >n2 with
xn3> xn2 .repeating this process leads to an infinite non
<…. decreasing subsequence xn1<xn2<xn3 As desired
Now suppose we have a bounded sequence in R by the
lemma there exists a monotone subsequence necessarily
bounded. But it follows from the monotone convergence
theorem that this subsequence must convergence and the
proof is complete ..
Finally the general case can be easily reduced to case of
n=1 as follows given a bounded sequence in R the
sequence of first coordinates is a bounded real sequence
hence has a convergent subsequence. We can then extract
a subsequence on which the second coordinates

15. converge and so on utill in the end we have passed from the
original sequence to a subsequence n times which is still a
subsequence on which each coordinate sequence converges
hence the subsequence itself is convergent..
Sequential compactness in euclidean spaces :
Suppose A is a subset of Rn with the property that every
sequence in A has a subsequence converging to an element
of A. then A must be bounded, since otherwise there exists a
sequence xm in A with ‖ xm ‖ > m for all m and then every
subsequence is unbounded and therefore not convergent.
Moreover A must be closed since from a non interior point x
in the complement of A one can build an A valued sequence
in A has a subsequence converging to an element of A .

16. Heine borel theorem 4:
In the topology of metric spaces the Heine boreal theorem
named after Eduard Heine and Emile boreal states :
For a subset S of Euclidean space Rn the following two
statements are equivalent :
. A is closed and bounded
. Every open cover of A has a finite sub-cover that is A is
compact.
proof : let A be a subset of R such that
A is closed & bounded
Let μ be a usual topology on Rn.
Then (Rn, μ ) is a topological space
Therefore (Rn, μ ) is a Hausdorff space
& every closed subset in a Hausdorff space is compact
therefore A is compact.
.

17. Conversly : suppose that A is compact subset in Rn
Therefore every compact subset In a Hausdorff space is
close
Therefore A is closed
It remains to prove that A is bounded x ϵ A &
Gx = A ∩ (x-1,x+1),Then ,A =U Gx
{Gx | x ϵ A } is a open cover of A
Since A is compact
Therefore it is covered by finite number of open sets
i.e AC UGxk
A C G x1 , U G x2, U…..UG xn
AC [A∩(x1-1, x1 +1)]U[A∩(x2 -1, x2 +1)]U…U[A∩(xn -1,xn +1)]
A C A∩[ (x1-1, x1 +1) U (x2 -1, x2 +1)U…..U(xn -1,xn +1)]
AC A∩[ {x1 x2 ……xn } -1, {x1 x2 ……xn } +1]
AC A∩[ min{x1 x2 ……xn } -1, max{x1 x2 ……xn } +1]
AC A∩ [m0-1,mo+1]
AC [m0-1,mo+1] hence A is bounded…..proved

18. Countably compact space :
In mathematics a topological space is countable compact if
every countable open cover has a finite sub-cover.
Sequentially compact space:
In mathematics a topological space is Sequentially compact
if every sequence has a convergent subsequence for general
topological spaces the notions of compactness and
sequential compactness are not equivalent they are however
equivalent for metric.
Definition (Local compactness) :
A topological space X is called locally compact at a point x X
if there exists a neighborhood Ux is compact .X is called
locally compact if X is compact at every point
An equivalent definition X is locally compact at x ϵ X iff there
is a closed set C c x such that C contains some
neighborhood of x.

19. idea of a metric space first suggested by M. frechet in
connection with the discussion of functn spaces. in turn out
that sets of objects of very different types carry natural metrics.
Totally bounded space :
In topological and related branches of mathematics a totally
bounded space that can be covered by finitely many subsets
of any fixed size ..the smaller the size fixed the more subsets
may be needed , but any specific size should require only
finitely many subsets .A related notion is totally bounded set in
which only a subset of space needs to be covered .every
subset of totally bounded space is a totally bounded set but
even if a space is not totally bounded some of its subsets still
will be.
Metric space:
A set X together with a metric ρ .the set theoretic approach

20. to the study of figures is based on the study of relative position
of their elementary constituents .A fundamental characteristic
of relative position of points of a point of a space is distance
between them .this approach leads to the As metric spaces
one may consider sets of sates function and mapping subsets
of Euclidean spaces and Hilbert spaces .metric are important
in the study o convergence and for the solution of questions
concerning approximation.
Complete metric spaces:
A metric space (X, ρ ) is called complete if each fundamental
sequence in it converges to a point of it the space (X, ρ T) is
always complete. completeness of a metric space is not a
topological property . A metric space homeomorphic to a
complete metric space may be non complete. For example
the real line R with the usual metric ρ (x,y)=| x-y |

21. Is homeomorphic to interval (0,1)= {x ϵ R :0<x<1} with the
same metric however the first metric space is complete
and the second is not.

22. REFERENCES :-
 Dr. H.K. Pathak ,(Topology)
 K.K .Jha , ( Advanced Gen. Topology)