This document introduces and investigates some weak separation axioms using the notion of πgb-closed sets. It defines πgb-closed sets, πgb-continuous functions, and various separation axioms including πgb-T0, πgb-T1, and πgb-T2. It introduces the concept of a πgb-D-set and defines associated properties like πgb-D0, πgb-D1, and πgb-D2 spaces. Results are proved relating these new concepts, showing properties like πgb-D1 spaces being πgb-T0 and πgb-D2 spaces being equivalent to
(𝛕𝐢, 𝛕𝐣)− RGB Closed Sets in Bitopological SpacesIOSR Journals
In this paper we introduce and study the concept of a new class of closed sets called (𝜏𝑖, 𝜏𝑗)− regular generalized b- closed sets (briefly(𝜏𝑖, 𝜏𝑗)− rgb-closed) in bitopological spaces.Further we define and study new neighborhood namely (𝜏𝑖, 𝜏𝑗)− rgb- neighbourhood (briefly(𝜏𝑖, 𝜏𝑗)− rgb-nhd) and discuss some of their properties in bitopological spaces. Also, we give some characterizations and applications of it.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
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.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
(𝛕𝐢, 𝛕𝐣)− RGB Closed Sets in Bitopological SpacesIOSR Journals
In this paper we introduce and study the concept of a new class of closed sets called (𝜏𝑖, 𝜏𝑗)− regular generalized b- closed sets (briefly(𝜏𝑖, 𝜏𝑗)− rgb-closed) in bitopological spaces.Further we define and study new neighborhood namely (𝜏𝑖, 𝜏𝑗)− rgb- neighbourhood (briefly(𝜏𝑖, 𝜏𝑗)− rgb-nhd) and discuss some of their properties in bitopological spaces. Also, we give some characterizations and applications of it.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
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.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
In this paper, we introduce the concepts of πgθ-closed map, πgθ-open map, πgθ-
homeomorphisms and πgθc-homeomorphisms and study their properties. Also, we discuss its relationship
with other types of functions.
Mathematics Subject Classification: 54E55
Fundamentals of Parameterised Covering Approximation SpaceYogeshIJTSRD
Combination of theories has not only advanced the research, but also helped in handling the issues of impreciseness in real life problems. The soft rough set has been defined by many authors by combining the theories of soft set and rough set. The concept Soft Covering Based Rough Set be given by J.Zhan et al 2008 , Feng Feng et al 2011 , S.Yuksel et al 2015 by taking full soft set instead of Covering. In this note We first consider the covering soft set and then covering based soft rough set. Again it defines a mapping from the coverings of element of universal set U to the parameters attributes . The new model “Parameterised Soft Rough Set on Covering Approximation Space is conceptualised to capture the issues of vagueness, and impreciseness of information. Also dependency on this new model and some properties be studied. Kedar Chandra Parida | Debadutta Mohanty "Fundamentals of Parameterised Covering Approximation Space" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-3 , April 2021, URL: https://www.ijtsrd.com/papers/ijtsrd38761.pdf Paper URL: https://www.ijtsrd.com/mathemetics/applied-mathematics/38761/fundamentals-of-parameterised-covering-approximation-space/kedar-chandra-parida
Abstract: In this paper, we define and study about a new type of generalized closed set called, g∗s-closed set.Its relationship with already defined generalized closed sets are also studied
In this paper, we introduce the concept of residual quotient of intuitionistic fuzzy subsets of ring and module and then define the notion of residual quotient intuitionistic fuzzy submodules , residual quotient intuitionistic fuzzy ideals. We study many properties of residual quotient relating to union, intersection, sum of intuitionistic fuzzy submodules (ideals). Using the concept of residual quotient, we investigate some important characterization of intuitionistic fuzzy annihilator of subsets of ring and module. We also study intuitionistic fuzzy prime submodules with the help of intuitionistic fuzzy annihilators. Many related properties are defined and discussed.
International Refereed Journal of Engineering and Science (IRJES) is a peer reviewed online journal for professionals and researchers in the field of computer science. The main aim is to resolve emerging and outstanding problems revealed by recent social and technological change. IJRES provides the platform for the researchers to present and evaluate their work from both theoretical and technical aspects and to share their views.
www.irjes.com
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
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.
A crystallographic group is a group acting on R^n that contains a translation subgroup Z^n as a finite index subgroup. Here we consider which Coxeter groups are crystallographic groups. We also expose the enumeration in dimension 2 and 3. Then we shortly give the principle under which the enumeration of N dimensional crystallographic groups is done.
In this paper, we introduce the concepts of πgθ-closed map, πgθ-open map, πgθ-
homeomorphisms and πgθc-homeomorphisms and study their properties. Also, we discuss its relationship
with other types of functions.
Mathematics Subject Classification: 54E55
Fundamentals of Parameterised Covering Approximation SpaceYogeshIJTSRD
Combination of theories has not only advanced the research, but also helped in handling the issues of impreciseness in real life problems. The soft rough set has been defined by many authors by combining the theories of soft set and rough set. The concept Soft Covering Based Rough Set be given by J.Zhan et al 2008 , Feng Feng et al 2011 , S.Yuksel et al 2015 by taking full soft set instead of Covering. In this note We first consider the covering soft set and then covering based soft rough set. Again it defines a mapping from the coverings of element of universal set U to the parameters attributes . The new model “Parameterised Soft Rough Set on Covering Approximation Space is conceptualised to capture the issues of vagueness, and impreciseness of information. Also dependency on this new model and some properties be studied. Kedar Chandra Parida | Debadutta Mohanty "Fundamentals of Parameterised Covering Approximation Space" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-3 , April 2021, URL: https://www.ijtsrd.com/papers/ijtsrd38761.pdf Paper URL: https://www.ijtsrd.com/mathemetics/applied-mathematics/38761/fundamentals-of-parameterised-covering-approximation-space/kedar-chandra-parida
Abstract: In this paper, we define and study about a new type of generalized closed set called, g∗s-closed set.Its relationship with already defined generalized closed sets are also studied
In this paper, we introduce the concept of residual quotient of intuitionistic fuzzy subsets of ring and module and then define the notion of residual quotient intuitionistic fuzzy submodules , residual quotient intuitionistic fuzzy ideals. We study many properties of residual quotient relating to union, intersection, sum of intuitionistic fuzzy submodules (ideals). Using the concept of residual quotient, we investigate some important characterization of intuitionistic fuzzy annihilator of subsets of ring and module. We also study intuitionistic fuzzy prime submodules with the help of intuitionistic fuzzy annihilators. Many related properties are defined and discussed.
International Refereed Journal of Engineering and Science (IRJES) is a peer reviewed online journal for professionals and researchers in the field of computer science. The main aim is to resolve emerging and outstanding problems revealed by recent social and technological change. IJRES provides the platform for the researchers to present and evaluate their work from both theoretical and technical aspects and to share their views.
www.irjes.com
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
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.
A crystallographic group is a group acting on R^n that contains a translation subgroup Z^n as a finite index subgroup. Here we consider which Coxeter groups are crystallographic groups. We also expose the enumeration in dimension 2 and 3. Then we shortly give the principle under which the enumeration of N dimensional crystallographic groups is done.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
In this paper, we define and study about a new type of generalized closed set called, 𝑔∗s-closed set.Its relationship with already defined generalized closed sets are also studied.
In this paper we introduce a new class of sets known as
δ
ˆ
S –closed sets in ideal topological
spaces and we studied some of its basic properties and characterizations. This new class of sets lies
between –I–closed [19] sets and g–closed sets, and its unique feature is it forms topology and it is
independent of open sets.
The aim of this paper is to study the class of β-normal spaces. The relationships among s-normal spaces, pnormal spaces and β-normal spaces are investigated. Moreover, we study the forms of generalized β-closed
functions. We obtain characterizations of β-normal spaces, properties of the forms of generalized β-closed
functions and preservation theorems.
The aim of this paper is to study the class of β-normal spaces. The relationships among s-normal spaces, pnormal spaces and β-normal spaces are investigated. Moreover, we study the forms of generalized β-closed
functions. We obtain characterizations of β-normal spaces, properties of the forms of generalized β-closed
functions and preservation theorems.
The aim of this paper is to study the class of β-normal spaces. The relationships among s-normal spaces, pnormal spaces and β-normal spaces are investigated. Moreover, we study the forms of generalized β-closed
functions. We obtain characterizations of β-normal spaces, properties of the forms of generalized β-closed
functions and preservation theorems.
The aim of this paper is to study the class of β-normal spaces. The relationships among s-normal spaces, pnormal spaces and β-normal spaces are investigated. Moreover, we study the forms of generalized β-closed
functions. We obtain characterizations of β-normal spaces, properties of the forms of generalized β-closed
functions and preservation theorems.
RW-CLOSED MAPS AND RW-OPEN MAPS IN TOPOLOGICAL SPACESEditor IJCATR
In this paper we introduce rw-closed map from a topological space X to a topological space Y as the image
of every closed set is rw-closed and also we prove that the composition of two rw-closed maps need not be rw-closed
map. We also obtain some properties of rw-closed maps.
On Fuzzy - Semi Open Sets and Fuzzy - Semi Closed Sets in Fuzzy Topologic...IOSR Journals
Abstract: The aim of this paper is to introduce the concept of fuzzy - semi open and fuzzy - semi closed sets of a fuzzy topological space. Some characterizations are discussed, examples are given and properties are established. Also, we define fuzzy - semi interior and fuzzy - semi closure operators. And we introduce fuzzy
- t-set, -SO extremely disconnected space analyse the relations between them.
MSC 2010: 54A40, 03E72.
Notions via β*-open sets in topological spacesIOSR Journals
In this paper, first we define β*-open sets and β*-interior in topological spaces.J.Antony Rex Rodrigo[3] has studied the topological properties of 𝜂 * -derived, 𝜂 * -border, 𝜂 * -frontier and 𝜂 * exterior of a set using the concept of 𝜂 * -open following M.Caldas,S.Jafari and T.Noiri[5]. By the same technique the concept of β*-derived, β*-border, β*-frontier and β*exterior of a set using the concept of β*-open sets are introduced.Some interesting results that shows the relationships between these concepts are brought about
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.
1. D.Sreeja, C.Janaki / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.031-037
On πgb-D-sets and Some Low Separation Axioms
D.Sreeja * and C.Janaki **
Asst.Professor, Dept. of Mathematics, CMS College of Science and Commerce
Coimbatore-6.
Asst.Professor, Dept. of Mathematics, L.R.G Govt.Arts College for Women, Tirupur-4.
Abstract
This paper introduces and investigates some (2) b-open [3] or sp-open [9], γ –open [11] if A ⊂
weak separation axioms by using the notions of πgb-
closed sets. Discussions has been carried out on its cl(int(A))∪int (cl(A)).
properties and its various characterizations. The complement of a b-open set is said to be
Mathematics Subject Classification: 54C05 b-closed [3]. The intersection of all b-closed sets of X
containing A is called the b-closure of A and is
denoted by bCl(A). The union of all b-open sets of X
Keywords: πgb-Ri, πgb-Di ,πgb-D-connected, πgb- contained in A is called b-interior of A and is denoted
D-compact.
by bInt(A). The family of all b-open (resp. α-open,
semi-open, preopen, β-open, b-closed, preclosed)
1.Introduction subsets of a space X is denoted by bO(X)(resp.
Levine [16] introduced the concept of αO(X), SO(X), PO(X), βO(X), bC(X), PC(X)) and
generalized closed sets in topological space and a the collection of all b-open subsets of X containing a
class of topological spaces called T ½ spaces. The fixed point x is denoted by bO(X,x). The sets SO(X,
investigation of generalized closed sets leads to x), αO(X, x), PO(X, x), βO(X, x) are defined
several new separation axioms. Andrijevic [3] analogously.
introduced a new class of generalized open sets in a Lemma 2.2 [3]: Let A be a subset of a space X. Then
topological space, the so-called b-open sets. This type
of sets was discussed by Ekici and Caldas [11] under (1) bCl(A) = sCl(A) ∩ pCl(A) = A∪[Int(Cl(A))
the name of -open sets. The class of b-open sets is ∩Cl(Int(A))];
contained in the class of semi-pre-open sets and (2) bInt(A) = sInt(A) ∪ pInt(A) = A∩ [Int(Cl(A)) ∪
contains all semi-open sets and pre-open sets. The Cl(Int(A))];
class of b-open sets generates the same topology as Definition 2.3 : A subset A of a space (X, τ) is called
the class of pre-open sets. Since the advent of these 1) a generalized b-closed (briefly gb-closed)[12] if
notions, several research paper with interesting bcl(A) ⊂ U whenever A ⊂ U and
results in different respects came to existence( U is open.
[1,3,7,11,12,20,21,22]). Extensive research on
generalizing closedness was done in recent years as 2) πg-closed [10] if cl(A)⊂U whenever A⊂ U and U
the notions of a generalized closed, generalized semi- is π-open.
closed, α-generalized closed, generalized semi-pre- 3) πgb -closed [23] if bcl(A)⊂U whenever A ⊂U and
open closed sets were investigated in [2,8,16,18,19]. U is π-open in (X, τ).
In this paper, we have introduced a new generalized By πGBC(τ) we mean the family of all πgb- closed
axiom called πgb-separation axioms. We have subsets of the space(X, τ).
incorporated πgb-Di, πgb-Ri spaces and a study has Definition 2.4: A function f: (X, τ) (Y, σ) is called
been made to characterize their fundamental 1) πgb- continuous [23] if every f-1(V) is πgb- closed
properties. in (X, τ) for every closed set V of (Y,σ).
2) πgb- irresolute [23] if f-1(V) is πgb- closed in(X, τ)
2. Preliminaries for every πgb -closed set V in (Y, σ).
Throughout this paper (X, τ) and (Y, τ) Definition[24]: (X, τ) is πgb -T0 if for each pair of
represent non-empty topological spaces on which no distinct points x, y of X, there exists a πgb -open set
separation axioms are assumed unless otherwise containing one of the points but not the other.
mentioned. For a subset A of a space (X,τ) , cl(A) Definition[24] : (X, τ) is πgb-T1 if for any pair of
and int(A) denote the closure of A and the interior of distinct points x, y of X, there is a πgb -open set U in
A respectively.(X, τ ) will be replaced by X if there is X such that x∈U and y∉U and there is a πgb -open
no chance of confusion.
Let us recall the following definitions which set V in X such that y∈U and x∉V.
we shall require later. Definition[24] : (X,τ) is πgb-T2 if for each pair of
Definition 2.1: A subset A of a space (X, τ) is called distinct points x and y in X, there exists a πgb -open
(1) a regular open set if A= int (cl(A)) and a regular set U and a πgb -open set V in X such that x∈U,
closed set if A= cl(int (A)); y∈Vand U∩V =Ф.
31 | P a g e
2. D.Sreeja, C.Janaki / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.031-037
Definition: A subset A of a topological space (X, τ ) is a πgb-D-set but not D-set,αD-set,pD-set,bD-set,sD-
is called: set.
(i) ) D-set [ 25] if there are two open sets U and V
such that U ≠X and A=U -V .
(ii) sD-set [5] if there are two semi-open sets U and V
such that U ≠X and A=U -V.
(iii) pD-set [14 ] if there are two preopen sets U and
V such that U ≠X and A=U -V .
(iv) αD-set [6] if there are two U,V∈αO(X,τ) such
that U≠X and A=U-V.
(v) bD-set [15] if there are two U,V∈BO(X,τ) such
that U≠X and A=U-V.
Definition 2.6[17]: A subset A of a topological space
X is called an g̃ αD-set if there are two g̃ α open sets
U,V such that U≠X and A=U-V.
Definition 2.7[4]: X is said to be (i)rgα-R0 iff rgα-
_
{x} ⊆G whenever x∈G∈RGαO(X). Definition 3.5: X is said to be
_ (i) πgb-D0 if for any pair of distinct points
(ii) rgα-R1 iff for x,y ∈X such that rgα- {x} ≠ rgα- x and y of X ,there exist a πgb-D-set in
_ X containing x but not y (or) a πgb-D-
{y} ,there exist disjoint U,V∈RGαO(X)such that set in X containing y but not x.
_ _ (ii) πgb-D1 if for any pair of distinct points
rgα- {x} ⊆U and rgα- {y} ⊆V. x and y in X,there exists a πgb-D-set of
Definition[13]: A topological space (X, τ) is said to X containing x but not y and a πgb-D-
be D-compact if every cover of X by D-sets has a set in X containing y but not x.
finite subcover. (iii) πgb-D2 if for any pair of distinct points
Definition[15]: A topological space (X, τ) is said to x and y of X,there exists disjoint πgb-D-
be bD-compact if every cover of X by bD-sets has a sets G and H in X containing x and y
finite subcover. respectively.
Definition[13]: A topological space (X, τ) is said to Example 3.6:Let X={a,b,c,d} and
be D-connected if (X, τ) cannot be expressed as the τ={Ф,{a},{a,b},{c,d},{a,c,d},X},then X is πgb-Di,
union of two disjoint non-empty D-sets. i=0,1,2.
Definition[15]: A topological space (X, τ) is said to Remark 3.7
be bD-connected if (X, τ) cannot be expressed as the (i) If (X, τ) is πgb-Ti, then (X, τ) is πgb-Di;i=0,1,2.
union of two disjoint non-empty bD-sets. (ii) If (X, τ) is πgb-Di,then it is πgb-Di-1;i=1,2.
(ii) If (X, τ) is πgb-Ti,then it is πgb-Ti-1;i=1,2.
3. πgb-D-sets and associated separation Theorem 3.8: For a topological space (X, τ).the
following statements hold.
axioms
(i) (X, τ) is πgb-D0 iff it is πgb-T0
Definition 3.1: A subset A of a topological space X
(ii) (X, τ) is πgb-D1 iff it is πgb-D2.
is called πgb-D-set if there are two U,V∈πGBO(X, τ)
Proof: (1)The sufficiency is stated in remark 3.7 (i)
such that U≠X and A=U-V.
Let (X, τ) be πgb-D0.Then for any two distinct points
Clearly every πgb-open set U different from X is a
πgb-D set if A=U and V=Ф. x,y∈X, atleast one of x,y say x belongs to πgbD-set G
Example 3.2: Let X={a,b,c} and where y∉G.Let G=U1-U2 where U1≠X and U1 and
τ={Ф,{a},{b},{a,b},X}.Then {c} is a πgb-D-set but U2∈πGBO(X, τ).Then x∈U1`.For y∉G we have two
not πgb-open.Since πGBO(X, τ)={ cases.(a)y ∉ U1 (b)y∈U1 and y ∈ U2.In case (a), x∈U1
Ф,{a},{b},{b,c},{a,c},{a,b},X}.Then U={b,c}≠X but y∉U1;In case (b); y ∈ U2 and x∉U2.Hence X is
and V={a,b} are πgb-open sets in X. For U and V, πgb-T0.
since U-V={b,c}-{a,b}={c},then we have S={c} is a (2) Sufficiency: Remark 3.7 (ii).
πgb-D-set but not πgb-open. Necessity: Suppose X is πgb-D1.Then for each
Theorem 3.3:Every D-set,αD-set,pD-set,bD-set,sD- distinct pair x,y∈X ,we have πgbD-sets G1 and G2
set is πgb-D-set. such that x∈G1 and y∉G1; x∉G2 and y∈G2. Let G1
Converse of the above statement need not be true as =U1-U2 andG2 =U3 –U4. By x∉G2, it follows that
shown in the following example. either x∉U3 or x∈ U3 and x ∈ U4
Example 3.4:Let X={a,b,c,d} and Now we have two cases(i)x ∉U3.By y ∉G, we have
τ={Ф,{a},{a,d},{a,b,d}{a,c,d}, two subcases (a)y ∉ U1.By x ∈ U1-U2,it follows that x
X}.πGBO(X,τ)=P(X).Hence πgb-D-set=P(X).{b,c,d} ∈U1-(U2∪ U3) and by y ∈ U3-U4,we have y ∈ U3-
32 | P a g e
3. D.Sreeja, C.Janaki / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.031-037
(U1∪ U4).Hence (U1-(U3∪ U4)) ∩ U3-(U1∪ U4)= Consequently, y ∈U ⊂({x})c.That is
c c
Ф.(b)y∈U1 and y∈U2 ,we have x∈ U1-U2 ({x}) =∪{U/y∈({x}) } which is πgb-open.
;y∈U2.⇒(U1-U2)∩U2=Ф. Conversely suppose {x} is πgb-closed for every x∈X.
Let x, y∈X with x≠y.Then x≠y ⇒y ∈ ({x})c.Hence
(ii)x∈U3 and x∈U4.We have y ∈U3-U4; x∈U4⇒(U3- ({x})c is a πgb-open set containing y but not x.
U4) ∩U4=Ф.Thus X is πgb-D2. Similarly ({y})c is a πgb-open set containing x but
Theorem 3.9: If (X, τ) is πgb-D1, then it is πgb-T0. not y. Hence X is πgb-T1-space.
Proof: Remark 3.7 and theorem 3.8 Corollary 3.18 : If X is πgb-T1, then it is πgb-
Definition 3.10: Let (X,τ) be a topological space. symmetric.
Let x be a point of X and G be a subset of X. Then Proof: In a πgb-T1 space, singleton sets are πgb-
G is called an πgb-neighbourhood of x (briefly πgb- closed. By theorem 3.17 , and by theorem 3.16 , the
nhd of x) if there exists an πgb-open set U of X space is πgb-symmetric.
such that x ∈U⊂G. Corollary 3.19: The following statements are
Definition 3.11: A point x∈X which has X as a πgb- equivalent
neighbourhood is called πgb-neat point. (i)X is πgb-symmetric and πgb-T0
Example (ii)X is πgb-T1.
3.12:LetX={a,b,c}.τ={Ф,{a},{b},{a,b},X}.πGBO(X, Proof: By corollary 3.18 and remark 3.7 ,it suffices
τ)={Ф,{a},{b},{a,b},{b,c},{a,c}, X}. The point {c} to prove (1) ⇒(2).Let x≠y and by πgb-T0 ,assume
is a πgb-neat point. that x∈G1 ⊂ ({y})c for some G1∈πGBO(X).Then x∉
Theorem 3.13: For a πgb-T0 topological space (X, τ), πgb-cl({y}) and hence y ∉ πgb-cl({x}).There exists a
the following are equivalent. G2 ∈ πGBO(X, τ) such that y∈G2⊂({x})c. Hence (X,
(i)(X, τ) is a πgb-D1 τ) is a πgb-T1 space.
(ii)(X, τ) has no πgb-neat point. Theorem 3.15: For a πgb-symmetric topological
Proof: (i) ⇒(ii).Since X is a πgb-D1,then each point x space(X, τ), the following are equivalent.
of X is contained in an πgb-D-set O=U-V and hence (1) X is πgb-T0
in U.By definition,U≠X.This implies x is not a πgb- (2) X is πgb-D1
neat point. (3) X is πgb-T1.
(ii)⇒(i) If X is πgb-T0,then for each distinct points x, Proof: (1)⇒(3):Corollary 3.19 and
y∈X, atleast one of them say(x) has a πgb- (3)⇒(2)⇒(1):Remark 3.7 .
neighbourhood U containing x and not y. Thus U≠X
is a πgbD-set. If X has no πgb-neat point, then y is
4. Applications
not a πgb-neat point. That is there exists πgb-
Theorem 4.1: If f : (X, τ)→(Y, σ) is a πgb-
neighbourhood V of y such that V ≠X. Thus y ∈ (V-
continuous surjective function and S is a D-set of
U) but not x and V-U is a πgb-D-set. Hence X is πgb-
(Y,σ ), then the inverse image of S is a πgb-D-set of
D 1.
(X, τ)
Remark 3.14 : It is clear that an πgb-T0 topological
Proof: Let U1 and U2 be two open sets of (Y, σ ).Let
space (X, τ)is not a πgb-D1 iff there is a πgb-neat
S=U1-U2 be a D-set and U1≠Y.We have f-1(U1)
point in X. It is unique because x and y are both πgb-
∊πGBO(X,τ) and f-1(U2) ∊πGBO(X, τ) and f-1(U1)
neat point in X, then atleast one of them say x has an
≠X.Hencef-1(S)=f-1(U1-U2)=f-1(U1)-f-1 (U2).Hence f-
πgb- neighbourhood U containing x but not y.This is 1
(S) is a πgb-D-set.
a contradiction since U ≠X.
Theorem 4.2 J: If f: (X, τ) → (Y, σ) is a πgb-
Definition 3.15: A topological space (X, τ) is πgb-
irresolute surjection and E is a πgb-D-set in Y,then
symmetric if for x and y in X, x ∈ πgb-cl({y}) ⇒y ∈
the inverse image of E is an πgb-D-set in X.
πgb-cl({x}).
Proof: Let E be a πgbD-set in Y.Then there are πgb-
Theorem 3.16: X is πgb-symmetric iff {x} is πgb-
open sets U1 and U2 in Y such that E= U1-U2 and
closed for x ∈X.
∪1≠Y.Since f is πgb-irresolute,f-1(U1) and f-1(U2) πgb-
Proof: Assume that x ∈ πgb-cl({y}) but y∉ πgb-
open in X.Since U1 ≠Y,we have f-1(U1) ≠X.Hence f-
cl({x}).This implies (πgb-cl({x})c contains y.Hence 1
(E) = f-1(U1-U2) = f-1(U1)- f-1(U2) is a πgbD-set.
the set {y} is a subset of (πgb-cl({x})c .This implies
Theorem 4.3: If (Y, σ) is a D1 space and f:(X, τ) →
πgb-cl({y}) is a subset of (πgb-cl({x})c. Now (πgb-
(Y, σ) is a πgb -continuous bijective function ,then
cl({x})c contains x which is a contradiction.
(X, τ) is a πgb-D1-space.
Conversely, Suppose that {x}⊂E∈πGBO(X, τ) but Proof: Suppose Y is a D1space.Let x and y be any
πgb-cl({y}) which is a subset of Ec and x ∉E. But this pair of distinct points in X,Since f is injective and Y
is a contradiction. is a D1space,yhen there exists D-sets Sx and Sy of Y
Theorem 3.17 : A topological space (X, τ) is a πgb- containing f(x) and f(y) respectively that f(x)∉Sy and
T1 iff the singletons are πgb-closed sets.
f(y)∉Sx. By theorem 4.1 f-1(Sx) and f-1(Sy) are πgb-D-
Proof:Let (X, τ) be πgb-T1 and x be any point of X.
sets in X containing x and y respectively such that
Suppose y ∈{X}c. Then x ≠y and so there exists a
x∉f-1(Sy) and y∉f-1(Sx).Hence X is a πgb-D1-space.
πgb-open set U such that y∈U but x ∉U.
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Vol. 2, Issue 5, September- October 2012, pp.031-037
Theorem 4.4: If Y is πgb-D1 and f: (X,τ)→ (Y,σ) is Definition 4.10: A topological space (X,τ) is said to
πgb-irresolute and bijective, then (X,τ) is πgb-D1. be πgb- D-compact if every cover of X by πgb-D-sets
Proof: Suppose Y is πgb-D1 and f is bijective, πgb- has a finite subcover.
irresolute.Let x,y be any pair of distinct points of Theorem 4.11:If a function f: (X,τ) → (Y,σ) is πgb-
X.Since f is injective and Y is πgb-D1,there exists continuous surjection and (X,τ) is πgb-D-compact
πgb-D-sets Gx and Gy of Y containing f(x) and f(y) then (Y, σ) is D-compact.
respectively such that f(y) ∉Gz and f(x) ∉Gy. By Proof:Let f: (X, τ) → (Y,σ) is πgb-continuous
theorem 4.2 , f-1(Gz) and f-1(Gy) are πgbD-sets in X surjection.Let {Ai:i∊∧} be a cover of Y by D-
containing x and y respectively.Hence X is πgb-D1. set.Then {f-1(Ai):i∊∧} is a cover of X by πgb-D-
Theorem 4.5: A topological space (X, τ) is a πgb-D1 set.Since X is πgb-D-compact,every cover of X by
if for each pair of distinct points x,y∊X,there exists a πgb-D set has a finite subcover ,say{f-1(A1),f-
1
πgb-continuous surjective function f: (X,τ) → (Y,σ) (A2)….f-1(An)}.Since f is onto,{A1,A2…….,An) is a
where (Y,σ ) is a D1 space such that f(x) and f(y) are cover of Y by D-set has a finite subcover.Therefore
distinct. Y is D-compact.
Proof: Let x and y be any pair of distinct points in Theorem 4.12: If a function f: (X,τ) → (Y,σ) is πgb-
X, By hypothesis,there exists a πgb-continuous irresolute surjection and (X, τ) is πgb-D-compact
surjective function f of a space (X, τ) onto a D1- then (Y, σ) is πgb-D-compact.
space(Y, σ )such that f(x)≠f(y).Hence there exists Proof:Let f: (X, τ) → (Y, σ) is πgb-
disjoint D-sets Sxand Sy in Y such that f(x)∊Sx and irresolutesurjection.Let {Ai:i∊∧} be a cover of Y by
f(y)∊Sy.Since f is πgb-continuous and surjective,by πgb-D-set.Hence Y= AiThen X=f-1(Y)=f-1( Ai)=
theorem 4.1 f-1(Sx) and f-1(Sy)are disjoint πgb-D-sets i i
in X containing x and y respectively.Hence (X, τ) is a f (Ai).Since f is πgb-irresolute,for each i∊∧,{f-
-1
πgb-D1-set. i
Theorem 4.6: X is πgb-D1 iff for each pair of 1
(Ai):i∊∧} is a cover of X by πgb-D-set.Since X is
distinct points x, y∈X ,there exists a πgb-irresolute πgb-D-compact,every cover of X by πgb-D set has a
surjective function f: (X,τ) → (Y,σ),where Y is πgb- finite subcover ,say{ f-1(A1), f-1(A2)…. f-1(An)}.Since
D1 space such that f(x) and f(y) are distinct. f is onto,{A1,A2…….,An) is a cover of Y by πgb-D-set
Proof: Necessity: For every pair of distinct points x, has a finite subcover.Therefore Y is πgb-D-compact.
y ∈X, it suffices to take the identity function on X.
Sufficiency: Let x ≠y ∈X.By hypothesis ,there exists 5.πgb-R0 spaces and πgb-R1 spaces
a πgb-irresolute, surjective function from X onto a Definition 5.1: Let (X, τ ) be a topological space then
πgb-D1 space such that f(x) ≠f(y).Hence there exists the πgb-closure of A denoted by πgb-cl (A) is defined
disjoint πgb-D sets Gx, Gy ⊂Y such that f(x) ∈Gx and by πgb-cl(A) = ∩ { F | F ∈ πGBC (X, τ ) and F⊃A}.
f(y) ∈Gy. Since f is πgb-irresolute and surjective, by Definition 5.2: Let x be a point of topological space
theorem 4.2 , f-1(Gx) and f-1(Gy) are disjoint πgb-D- X.Then πgb-Kernel of x is defined and denoted by
sets in X containing x and y respectively. Therefore
X is πgb-D1 space. Ker πgb{x}=∩{U:U∈πGBO(X) and x∈U}.
Definition 4.7: A topological space (X, τ) is said to Definition 5.3: Let F be a subset of a topological
be πgb-D-connected if (X, τ) cannot be expressed as space X .Then πgb-Kernel of F is defined and
the union of two disjoint non-empty πgb-D-sets. denoted by Ker πgb(F)=∩{U:U∈πGBO(X) and
Theorem 4.8: If (X, τ) → (Y, σ) is πgb-continuous F⊂U}.
surjection and (X, τ) is πgb-D-connected, then (Y, σ) Lemma 5.4: Let (X, τ) be a topological space and x
is D-connected.
Proof: Suppose Y is not D-connected. Let Y=A∪B ∈ X.Then Ker πgb(A) = {x ∈X| πgb-cl({x})∩A ≠Φ}.
where A and B are two disjoint non empty D sets in Proof:Let x ∈ Kerπgb(A) and πgb-cl({x})∩A = Φ.
Y. Since f is πgb-continuous and onto, X=f-1(A)∪f- Hence x ∉X−πgb-cl({x}) which is an πgb-open set
1
(B) where f-1(A) and f-1(B) are disjoint non-empty containing A. This is impossible, since x ∈ Kerπgb
πgb-D-sets in X. This contradicts the fact that X is (A).
πgb-D-connected. Hence Y is D-connected. Consequently, πgb- cl({x})∩A ≠Φ Let πgb- cl({x})∩
Theorem 4.9: If (X,τ) → (Y,σ) is πgb-irresolute A ≠Φ and x ∉Ker πgb (A). Then there exists an πgb-
surjection and (X, τ) is πgb-D-connected, then (Y, σ )
is πgb-D-connected. open set G containing A and x ∉G. Let y ∈ πgb-
Proof: Suppose Y is not πgb-D-connected.Let cl({x})∩A. Hence G is an πgb- neighbourhood of y
Y=A∪B where A and B are two disjoint non empty where x ∉G. By this contradiction, x ∈ Ker πgb(A).
πgb-D- sets in Y.Since f is πgb-irreesolute and Lemma 5.5: Let (X, τ) be a topological space and x
onto,X=f-1(A)∪f-1(B) where f-1(A) and f-1(B) are ∈ X. Then y ∈ Kerπgb({x}) if and only if x ∈ πgb-
disjoint non-empty πgb-D-sets in X.This contradicts Cl({y}).
the fact that X is πgb-D-connected. Hence Y is πgb-
D-connected.
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Vol. 2, Issue 5, September- October 2012, pp.031-037
Proof: Suppose that y ∉ Ker πgb({x}). Then there (iv) Each πgb-closed F can be expressed as F=∩{G:G
exists an πgb-open set V containing x such that y ∉V. is πgb-open and F⊂G}
Therefore we have x ∉πgb-cl({y}). Converse part is (v) ) Each πgb-open G can be expressed as G=∪{A:A
similar. is πgb-closed and A⊂G}
Lemma 5.6: The following statements are equivalent (vi) For each πgb-closed set,x∉F implies πgb-
for any two points x and y in a cl{x}∩F=Φ.
topological space (X, τ) : Proof: (i)⇒(ii): For any x ∈X, we have Ker πgb{x}=
(1) Ker πgb ({x}) ≠Ker πgb({y}); ∩{U:U∈πGBO(X). Since X is πgb-R0there exists
(2) πgb- cl({x}) ≠ πgb-cl({y}). πgb-open set containing x contains πgb-cl{x}.Hence
Proof: (1) ⇒ (2): Suppose that Ker πgb({x}) ≠ Ker πgb- cl{x}⊂Ker πgb{x}.
πgb({y}) then there exists a point z in X such that z
(ii) ⇒ (iii): Let x ∉F∈πGBC(X). Then for any y ∈F,
πgb-cl{y}⊂F and so x∉πgb- cl{y}⇒y∉πgb-
∈X such that z∈Ker πgb({x}) and z ∉ Ker πgb({y}). It
cl{x}.That is there exists Uy∈πGBO(X) such that y∈
follows from z ∈ Ker πgb({x}) that {x} ∩ πgb-cl({z}) Uy and x∉Uy for all y∈F. Let U=∪{Uy∈πGBO(X)
≠Φ .This implies that x ∈ πgb-cl({z}). By z ∉Ker such that y∈ Uy and x ∉ Uy}.Then U is πgb-open such
that x∉U and F⊂U.
πgb({y}), we have {y}∩πgb- cl({z}) = Φ. Since x ∈
(iii)⇒(iv): Let F be any πgb-closed set and N=∩{G:G
πgb-cl({z}) , πgb-cl({x}) ⊂ πgb-cl({z}) and {y} ∩ is πgb-open and F⊂G}.Then F⊂N---(1).Let x∉F,then
πgb-cl({z}) = Φ. Therefore, πgb-cl({x}) ≠ πgb- by (iii) there exists G∈πGBO(X) such that x∉G and
cl({y}). Now Ker πgb({x})≠ Ker πgb({y}) implies that F⊂G, hence x∉N which implies x∈N ⇒x∈F. Hence
πgb-cl({x}) ≠ πgb-cl({y}). N⊂F.---(2).From (1) and (2),each πgb-closed
(2) ⇒ (1): Suppose that πgb-cl({x}) ≠ πgb-cl({y}). F=∩{G:G is πgb-open and F⊂G}.
Then there exists a point z ∈ X such that z ∈ πgb- (iv)⇒(v) Obvious.
cl({x}) and z ∉ πgb-cl({y}). Then, there exists an (v) ⇒(vi) Let x∉F∈πGBC(X).Then X-F=G is a πgb-
πgb-open set containing z and hence containing x but open set containing x.Then by (v),G can be expressed
not y, i.e., y ∉ Ker({x}). Hence Ker({x}) ≠Ker({y}). as the union of πgb-closed sets A ⊆G and so there is
Definition5.7: A topological space X is said to be an M∈πGBC(X) such that x∈M⊂G and hence πgb-
cl{x}⊂G implies πgb- cl{x}∩F=Φ.
πgb-R0 iff πgb-cl{x}⊆G whenever x∈G∈ πGBO(X). (vi)⇒(i) Let x ∈G∈πGBO(X).Then x∉(X-G) which is
Definition5.8: A topological space (X,τ) is said to be πgb-closed set. By (vi) πgb- cl{x}.∩(X-G)=Φ.⇒πgb-
πgb-R1 if for any x,y in X with πgb-cl({x})≠πgb- cl{x}⊂G.Thus X is πgb-R0-space.
cl({y}),there exists disjoint πgb-open sets U and V Theorem 5.12 :A topological space (X, τ) is an πgb-
such that πgb-cl({x})⊆U and πgb-cl({y})⊆V R0 space if and only if for any x and y in X, πgb-
Example 5.9: Let cl({x})≠πgb-cl({y}) implies πgb-cl({x}) ∩πgb-
X={a,b,c,d}.τ={Ф,{b},{a,b},{b,c},{a,b,c},X}.πGBO cl({y}) =Φ.
(X,τ)=P(X)Then X is πgb-R0 and πgb-R1. Proof: Necessity. Suppose that (X, τ ) is πgb-R and
Theorem 5.10 : X is πgb-R0 iff given x≠y∈; πgb- x, y ∈ X such that πgb-cl({x}) ≠πgb-cl({y}). Then,
cl{x}≠ πgb-cl{y}. there exist z ∈πgb-cl({x}) such that z∉πgb-cl({y})
Proof: Let X be πgb-R0 and let x≠y∈X.Suppose U is (or z ∈cl({y})) such that z ∉πgb-cl({x}). There
exists V ∈πGBO(X) such that y ∉ V and z ∈V
a πgb-open set containing x but not y, then y∈ πgb-
.Hence x ∈ V . Therefore, we have x ∉πgb-cl({y}).
cl{y}⊂X-U and hence x∉ πgb-cl{y}.Hence πgb- Thus x ∈ (πgb-cl({y}) )c∈πGBO(X ), which implies
cl{x}≠ πgb-cl{y}. πgb-cl({x}) ⊂(πgb-cl({y}) )cand πgb-cl({x}) ∩πgb-
Conversely,let x≠y∈X such that πgb-cl{x}≠ πgb- cl({y}) =Φ.
Sufficiency. Let V ∈πGBO(X) and let x ∈V .To show
cl{y},.This implies πgb-cl{x}⊂X- πgb-
that πgb-cl_({x}) ⊂V. Let y ∉V , i.e., y ∈Vc. Then x
cl{y}=U(say),a πgb-open set in X. This is true for
≠y and x ∉πgb-cl({y}). This showsthat πgb-cl({x})
every πgb-cl{x}.Thus ∩πgb-cl{x}⊆U where x∈ ≠πgb-cl({y}). By assumption ,πgb-cl({x}) ∩πgb-
πgb-cl{x}⊂U∈ πGBO(X).This implies ∩πgb- cl({y}) =Φ. Hencey ∉πgb-cl({x}) and therefore πgb-
cl{x}⊆U where x∈ U∈ πGBO(X).Hence X is πgb- cl({x}) ⊂ V .
R0. Theorem 5.13 : A topological space (X, τ ) is an
Theorem 5.11 : The following statements are πgb- R0 space if and only if for anypoints x and y in
equivalent X , Ker πgb({x}) ≠Ker πgb({y}) implies Ker πgb({x})
(i)X is πgb-R0-space ∩Ker πgb({y})=Φ.
(ii)For each x∈X,πgb-cl{x}⊂Kerπgb{x} Proof. Suppose that (X, τ) is an πgb- R0 space. Thus
by Lemma 5.6, for any points x and y in X if
(iii)For any πgb-closed set F and a point x∉F,there
Kerπgb({x})≠Kerπgb({y}) then πgb-cl({x})≠πgb-
exists U∈πGBO(X) such that x∉U and F⊂U,
cl({y}).Now to prove that Kerπgb({x})∩Kerπgb({y})
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=Φ. Assume that z ∈Ker πgb({x})∩Ker πgb({y}). By z Kerπgb({x}).Since x ∈ πgb-cl({x}) and πgb-cl({x}) is
∈ Kerπgb({x}) and Lemma 5.5, it follows that x∈πgb- πgb-closed,then by (iv) we get y ∈ Kerπgb({x})⊆ πgb-
cl({z}). Since x∈πgb-cl({z}); πgb-cl({x})=πgb- cl({x}),Therefore x ∈ πgb-cl({y}) ⇒y ∈ πgb-cl({x}).
cl({z}). Similarly, we have πgb-cl({y}) = πgb- Conversely, let y∈πgb-cl({x}).By lemma 5.5, x
cl({z})=πgb-cl({x}). This is a contradiction. ∈Kerπgb({y}).Since y∈πgb-cl({y})and πgb-cl({y}) is
Therefore, we have Ker πgb({x}) ∩Ker πgb({y})=Φ πgb-closed,then by (iv) we get x∈Kerπgb({y})⊆πgb-
Conversely, let (X, τ) be a topological space such that cl({y}).Thus y∈πgb-cl({x})⇒x∈πgb-cl({y}).By
for any points x and y inX such that πgb-cl{x}≠πgb- theorem 5.14, we prove that (X,τ) is πgb-R0 space.
cl{y}.Kerπgb({x}) ≠Kerπgb({y}) implies Ker πgb({x}) Remark 5.17: Every πgb-R1 space is πgb-R0 space.
∩Ker πgb({y})=Φ.Since z∈πgb-cl{x}⇒x∈Ker πgb({z}) Let U be a πgb-open set such that x∈U. If y∉U,then
and therefore Ker πgb({x}) ∩Ker πgb({y})≠Φ.By since x∉πgb-cl({y}),πgb-cl({x})≠πgb-cl({y}).Hence
hypothesis,we have Kerπgb({x})=Ker πgb({z}). Then z there exists an πgb-open set V such that y∈V such
∈πgb-cl({x}) ∩πgb-cl({y}) implies that Kerπgb({x}) = that πgb-cl({y})⊂V and x∉V⇒y∉πgb-cl({x}).Hence
Kerπgb({z})=Kerπgb({y}). This is a contradiction. πgb-cl({x})⊆U. Hence (X,τ) is πgb-R0.
Hence πgb-cl({x})∩πgb-cl({y})=Φ;By theorem Theorem 5.18: A topological space (X, τ) is πgb-R1
5.12, (X, τ) is an πgb-R0 space. iff for x,y∈X, Kerπgb({x})≠πgb-cl({y}),there exists
Theorem 5.14 : For a topological space (X, τ ), the disjoint πgb-open sets U and V such that πgb-
following properties are equivalent. cl({x})⊂ andπgb-cl({y}) ⊂V.
(1) (X, τ) is an πgb-R0 space Proof: It follows from lemma 5.5.
(2) x∈πgb-cl({y}) if and only if y∈πgb-cl({x}), for
any points x and y in X. References
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∈πgb-cl({x}) and hence Kerπgb({x}) ⊂πgb-cl({x}). operator,Bol.Soc.Paran.Mat.(3s) v.21 ½
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