Download τ* -Generalized Compact Spaces and τ* -Generalized

S.Eswaran et. al. / International Journal of Engineering Science and Technology
Vol. 2(5), 2010, 2466-2469
*-Generalized Compact Spaces and
*-Generalized Connected Spaces
in Topological Spaces
1
S.ESWARAN*
2
A.PUSHPALATHA
1
Department of Mathematics, KPR Institute of Engineering and Technology,
Coimbatore, Tamilnadu, India – 641 407
2
Department of Mathematics, Government Arts College,
Udumelpet, Tamilnadu, India – 642 126
1
[email protected]
2
[email protected]
Abstract
In this paper, we introduce new topological spaces called *-generalized compact spaces and *-generalized
connected spaces using *-generalized open sets and study some of their properties.
2000 Mathematics Subject Classification: 54A05.
Key words: *-g-compact spaces, *-g-connected spaces.
1. Introduction
In 1976, Thompson [9] introduced the notion of S-closed spaces by using semi-open sets. DI Maio and Noiri [2]
defined and studied about a class of compact space called s-closed space using semi-open cover. As a generalization
of S-closed space, Noiri [6] introduced and investigated the notion of locally S-closed spaces. Balachandran,
Sundaram and Maki [1] introduced a class of compact space called GO-compact space and GO-connected space
using g-open cover. Pushpalatha, Eswaran and Rajarubi [7] introduced and studied *-generalized closed sets and *generalized open sets. Using *-generalized closed sets, Eswaran and Pushpalatha [4] introduced *-generalized
continuous maps. In [8], Pushpalatha and Eswaran studied some strong forms of *-generalized continuous maps.
In this paper, we introduce new classes of topological spaces called *-generalized compact spaces and *generalized connected spaces using *-generalized open sets.
Throughout this paper (X, τ*) and (Y, *) (or simply X and Y) represent non-empty topological spaces on
which no separation axioms are assumed, unless otherwise mentioned. For a subset A of a space (X, τ*), cl*(A), cl
(A) and Ac represent generalized closure operator, closure of A and complement of A respectively.
2. Preliminaries
We recall the following definitions.
Definition 2.1. A subset A of a topological space (X, ) is called generalized closed[5] (briefly g-closed) in X
if cl(A)  G whenever A  G and G is open in X. A subset A is called generalized open (briefly g-open) in X if AC
is g-closed.
Definition 2.2. For the subset A of a topological space X,
(i) the generalized closure operator cl*[3] is defined by the intersection of all g- closed sets containing A.
(ii) the topology *[3] is defined by * ={G : cl*(GC) = GC }
ISSN: 0975-5462
2466
S.Eswaran et. al. / International Journal of Engineering Science and Technology
Vol. 2(5), 2010, 2466-2469
Definition 2.3. A subset A of a topological space X is called *-generalized closed set [7] (briefly *-g-closed) if
*-generalized closed set is called the
cl (A)  G whenever A  G and G is *-open. The complement of
*
*
 -generalized open set (briefly  -g-open).
Definition 2.4. A topological space (X, τ*) is called τ*-Tg [4] space if every τ*-g-closed set in X is g-closed in
X.
Definition 2.5. A map f : X  Y from a topological space (X, τ*) into a topological space (Y, σ*) is
called
(i) τ*-generalized continuous [4] (briefly
τ*-g-continuous) if the inverse image of every g-closed (or g-open)
set in Y is
τ*-g-closed (or τ*-g-open) in X.
(ii) strongly τ*- generalized continuous [8] (briefly strongly τ*-g-continuous) if the inverse image of every τ*-gopen set (or τ*-g-closed set) in Y is g-open (or g-closed) in X.
(iii) τ*-gc-irresolute [8] if the inverse image of every τ*-g-closed set in Y is τ*-g-closed in X.
*
3. *-Generalized Compact Spaces
In this section, we introduce and study a class of compact space called *-Generalized Compact Spaces in
topological spaces.
Definition 3.1. A collection {Ai : i  I} of τ*-g-open sets in a topological space (X, τ*) is called a τ*-g-open
cover of a subset B if B
 {Ai : i  I}.
Definition 3.2. A topological space (X, τ*) is called τ*-generalized compact (briefly
τ*-g-compact), if
every τ*-g-open cover of X has a finite subcover.
Definition 3.3. A subset B of a topological space (X, τ*) is said to be τ*-g-compact relative to X , if for every
τ*-g-open subsets of X such that B   {Ai : i  I} there exists a finite subset Io of I
collection {Ai : i  I} of
such that
B   {Ai : i  Io}.
Definition 3.4. A subset B of a topological space (X, τ*) is said to be τ*-g-compact if B is τ*-g-compact as the
subset of X.
Theorem 3.5. A τ*-g-closed subset of
τ*-g-compact space is τ*-g-compact relative to X.
Proof : Let A be a τ*-g-closed subset of a τ*-g-compact space X. Then AC is τ*-g-open in X. Let S be a cover
of A by τ*-g-open sets in X. Then, {S, AC} is a τ*-g-open cover of X. Since X is τ*-g-compact, it has a finite
subcover, say {G1, G2, ., Gn}. If this subcover contains AC, we discard it. Otherwise leave the subcover as it is. Thus
we have obtained a finite τ*-g-open subcover of A and so A is τ*-g-compact relative to X.
Theorem 3.6. (i) A τ*-g-continuous image of a τ*-g-compact space is compact. (ii) If a map f : X  Y is
τ*-gc-irresolute and a subset B is τ*-g-compact relative to X, then the image f(B) is τ*-g-compact relative to Y.
Proof : (i) Let f : X  Y be a
τ*-g-continuous map from a τ*-g-compact space X onto a
topological space Y. Let {Ai : i  I} be an open cover of Y. Then {f -1(Ai) : i  I} is a τ*-g-open cover of X. Since
X is τ*-g-compact, it has a finite subcover, say { f -1(A1), f -1(A2), …, f -1(An)}. Since f is onto, {A1, A2,…,An} is a
g-open cover of Y and so Y is compact.
(ii) Let {Ai : i  I} be any collection of
τ*-g-open subsets of Y such that f(B)   {Ai : i  I}. Then, B  
{f -1(Ai) : i  I}. By using assumptions there exists a finite subset Io of I such that B {f -1(Ai) : i  I}. Therefore, we
have f(B)   {Ai : i  Io} which shows that f(B) is τ*-g-compact relative to Y.
Theorem 3.7. If f : X  Y is a strongly τ*-g-continuous map from a compact space X onto a topological
space Y, then Y is
τ*-g-compact.
Proof : Let {Ai : i  I} be a τ*-g-open cover of Y. Then {f -1(Ai) : i  I} is a g-open cover of X since f is
f -1(A2) , ..,
f -1(An)}.
strongly τ*-g-continuous. Since X is compact, it has a finite subcover say {f -1(A1),
Since f is onto,{A1, A2, ., An} is a finite τ*-g-open cover of Y. Therefore Y is τ*-g-compact.
4. τ*-Generalized Connectedness.
Here we introduce a new class of spaces called τ*-generalized connected spaces and prove some of its
properties.
Definition 4.1. A topological space X is said to be τ*-generalized connected (briefly
τ*-gconnected) if X cannot be written as a disjoint union of two nonempty τ*-g-open sets. A subset of X is τ*-gconnected if it is
τ*-g-connected as a subspace.
Theorem 4.2. For a topological space X, the following are equivalent:
(a) X is τ*-g-connected
ISSN: 0975-5462
2467
S.Eswaran et. al. / International Journal of Engineering Science and Technology
Vol. 2(5), 2010, 2466-2469
(b) The only subsets of X which are both τ*-g-open and τ*-g-closed are the empty set  and X.
(c) Each τ*-g-continuous map of X into a discrete space Y with at least two points is a constant map.
Proof : (a)  (b) Let U be a τ*-g-open and τ*-g-closed subset of X. Then X – U is both
τ*-g-open and
τ*-g-closed. Since X is the disjoint union of the τ*-g-open sets U and X – U, one of these must be empty, that is
U =  or U = X.
(b)  (a) Suppose that X = A  B where A and B are disjoint nonempty τ*-g-open subsets of X. Then A is
both τ*-g-open and τ*-g-closed. By assumption, A =  or X. Therefore X is
τ*-g-connected.
τ *-g-continuous map. Then X is covered by τ*-g-open and τ*(b)  (c) Let f : X  Y be a
g-closed covering {f -1(y) : y  Y}. By assumption, f -1(y) =  or X for each y Y. If f -1(y) =  for all y  Y then f
fails to be a map. Then, there exists only one point y  Y such that f -1(y)   and hence f -1(y) = X which shows
that f is a constant map.
(c)  (b) let U be both τ*-g-open and
τ*-g-closed in X. Suppose U  . Let f : X  Y be a τ*-gcontinuous map defined by f(U) = {y} and f(X – U) = {w} for some distinct points y and w in Y. By assumption, f is
constant. Therefore, y = w and so U = X.
Theorem 4.3. Every τ*-g-connected space is connected.
Proof : Let X be a τ*-g-connected space. If possible let X be not connected. Then X can be written as X = A  B
where A and B are disjoint nonempty g-open sets in X. In [7] it has been proved in Theorem 3.4 that every g-open
set is τ*-g-open. Thus, X = A  B where A and B are disjoint, are nonempty and τ*-g-open sets in X. This
contradicts the fact that X is τ*-g-connected. Therefore X is connected.
Theorem 4.4. Suppose that X is a τ*-Tg topological space. Then X is connected if and only if X is τ*-gconnected.
τ*-g-connected. Then X can
Proof: Assume that X is τ*-Tg and connected. If possible, let X be not
be written in the form X = A  B where A and B are nonempty, disjoint and τ*-g-open sets in X. Since X is τ*-Tg,
every τ*-g-open set is g-open and so X = A  B where A and B are nonempty, disjoint and g-open sets in X. This
contradicts the fact that X is connected. Therefore X is τ*-g-connected.
Converse follows from Theorem 4.3.
τ*-g-continuous surjection and X is
τ*-g-connected then
Theorem 4.5. (i) If f : X  Y is
Y is connected.
(ii) If f : X  Y is a τ*-gc-irresolute surjection and X is τ*-g-connected then Y is
τ*-gconnected.
Proof: (i) suppose that Y is not connected. Let Y = A  B where A and B are disjoint nonempty τ*-open sets in
Y. Since f is
τ*-g-continuous and onto, X = f -1(A)  f -1(B) where f -1(A) and f -1(B) are disjoint nonempty
τ*-g-open sets in X. This contracts the fact that X is τ*-g-connected. Hence Y is connected.
(ii) If possible, assume that Y is not
τ*-g-connected. Then Y = A  B where A and B are nonempty,
f -1(B) are τ*-g-open in X. Since
disjoint and τ*-g-open sets in Y. Since f is τ*-gc-irresolute, where f -1(A) and
-1
-1
-1
-1
-1
f (A  B) = f (A)  f -1(B). Thus X is a union
f is onto, f (A) and f (B) are nonempty. Now X = f (Y) =
of disjoint, nonempty and τ*-g-open sets in X. This contradicts the fact that X is
τ*-g-connected.
Therefore, Y is τ*-g-connected.
With reference to Theorem 4.5.(i) we have the following theorem.
Theorem 4.6. Let f be a τ*-g-continuous map from a topological space (X, τ*) into a topological space (Y, σ*).
Then f(H) is a connected subset of Y for every g-closed and τ*-g-connected subset H of X.
Proof : In [4], it has been proved in Theorem 3.13 that the restriction f/H of f to H is
τ*-g-continuous.
By Theorem 4.5, the image of the τ*-g-connected space (H, τ*/H) under f / H : ( H , * / H )   * / f ( H ) is connected.
Thus (f(H), σ*/f(H)) is connected. Therefore, f(H) is a connected subset of Y.
Theorem 4.7. If f : X  Y is a strongly τ*-g-continuous map from a connected space X onto a topological
space Y then Y is
τ*-g-connected
Proof : If possible, let Y be not
τ*-g-connected. Then Y can be written in the form Y = A  B where
A and B are disjoint nonempty τ*-g-open sets in Y. Since f is strongly τ*-g-continuous, f -1(A) and f -1(B) are g-open
sets in X. Also X = f -1(Y) = f -1(A  B) = f -1(A) 
f -1(B). This contradicts the fact that X is connected. Therefore
Y is τ*-g-connected.
References
ISSN: 0975-5462
2468
S.Eswaran et. al. / International Journal of Engineering Science and Technology
Vol. 2(5), 2010, 2466-2469
[1] K.Balachandran, P.Sundaram and J.Maki, On generalized continuous maps in topological spaces, Mem. Fac. Sci. Kochi Univ. (Math.) 12
(1991), 5-13.
[2] G. Di Maio and T.Noiri, Indian J. pure appl. Math., 18(1987), 226 – 33.
[3] W.Dunham, A new closure operator for non-T topologies, Kyungpook Math. J., 22 (1982), 55 – 60
[4] S.Eswaran and A.Pushpalatha, *-generalized continuous maps in topological spaces, International J. of Math Sci & Engg. Appls. (IJMSEA)
ISSN 0973-9424 Vol. 3, No. IV, (2009), pp. 67 – 76
[5] N.Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo(2), 19 (1970), 89 – 96.
[6] T.Noiri, Atti Accad. Naz. Lincei Rent. Cl. Sci. Fis. Mat., Natur. (8) 64 (1978), 157 – 62.
[7] A.Pushpalatha, S.Eswaran and P.Rajarubi, *-generalized closed sets in topological spaces, Proceedings of World Congress on Engineering
2009 Vol II WCE 2009, July 1 – 3, 2009, London, U.K., 1115 – 1117
[8] A.Pushpalatha and S.Eswaran, Strong forms of *-generalized continuous maps in topological spaces, Int. J. Contemp. Math. Sciences, Vol. 5,
2010, no. 17, 815 - 822,
[9] T.Thompson, Proc. Am. Math. Soc. 60(1976), 335 – 38
ISSN: 0975-5462
2469