Volume 24, No. 1, March 2011
APPLICATION OF eGα-CLOSED SETS
Young Key Kim*, R. Devi**, and A. Selvakumar***
Abstract. The notion of egα− closed sets in a topological space introduced by R. Devi and A. selvakumar [2]. In this paper, we introduce the concept of egα-US spaces by utilizing egα − open sets and study the basic properties of this space.
1. Introduction
In 1967, A. Wilansky [7] introduced and studied the concept of U S- spaces. Also, the notion of egα-closed sets of a topological space are discussed by R. Devi et. al. [2]. The concept of slightly continuous functions are introduced and investigated by R. C. Jain [4].
The aim of this paper is to introduce the notion of slightly egα- continuous functions and egα-US spaces. Further, the basic properties of slightly egα-continuous functions are derived. Also we studied the concepts ofegα-spaces, egα-convergence, sequentially egα-compactness, se- quentially egα-continuity and sequentially egα-sub-continuity.
Throughout the present paper, X and Y are always topological spaces.
Let A be a subset of X. We denote the interior and the closure of a set A by int(A) and cl(A) respectively.
A subset A of a space X is said to be α-open [5] if A⊆ int(cl(int(A))).
A subset A of a space X is said to be egα-closed [2] if αcl(A) ⊆ U whenever A ⊆ U and U is α-open. The complement of a egα-closed set is said to be egα-open. The intersection of all egα-closed sets of X containing A is called egα-closure of A and is denoted by egαcl(A). The union of all egα-open sets of X contained in A is called egα-interior of A and is denoted by egαint(A).
Received June 18, 2010; Revised November 09, 2010; Accepted December 13, 2010.
2010 Mathematics Subject Classification: Primary 54A08.
Key words and phrases: egα − open set, egα − US space.
Correspondence should be addressed to Young Key Kim, [email protected].
The family of all α-open (resp. egα-open, egα-closed, clopen, egα- clopen) set of X is denoted by αO(X) (resp. egαO(X), egαC(X), CO(X), egαCO(X)).
Definition 1.1. ([2]) A function f : X → Y is egα-continuous if f−1(V ) is egα-open set in X for each open set V of Y .
Definition 1.2. ([4]) A function f : X → Y is slightly- continuous if f−1(V ) is open set in X for each clopen set V of Y .
2. eGα-U S spaces
Definition 2.1. A sequence {xn} in a space X, egα-converges to a point x∈ X if {xn} is eventually in every egα-open set containing x.
Definition 2.2. A space X is said to be egα-U S if every sequence in X, egα-converges to a point of X.
Definition 2.3. A space X is said to be
(i) egα-T1 if each pair of distinct points x and y in X there exists a egα-open set U in X such that x ∈ Uand y /∈ U and a egα-open set V in X such that y ∈ V and x /∈ V .
(ii) egα-T2 if for each pair of distinct points x and y in X there exist egα-open sets U and V such that U ∩ V = ϕ and x ∈ U, y ∈ V . Theorem 2.4. Every egα-U S-space is egα-T1.
Proof. Let X be a egα-US-space and x, y be two distinct points of X. Consider the sequence {xn}, where xn = x for any n∈ N. Clearly {xn} egα-converges to x. Since x ̸= y and X is egα-US, {xn} does not egα-converges to y, i.e., there exists a egα-open set U containing x but not y. Similarly, we obtain a egα-open set V containing y but not x. Thus, X is egα-T1.
Theorem 2.5. Every egα-T2 space is egα-US.
Proof. Let X be a egα-T2 space and {xn} a sequence in X. Assume that {xn} egα-converges to two distinct points x and y. Then {xn} is eventually in everyegα-T2then{xn} is eventually in two disjoint egα-open sets. This is a contradiction. Therefore, X is egα-US.
Definition 2.6. A subset A of a space X is said to be
(i) sequentially egα-closed if every sequence in A egα-converges to a point in A,
(ii) sequentially eGαO-compact if every sequence in A has a subse- quence which egα-converges to a point in A.
Theorem 2.7. A space is egα-U S if and only if the diagonal set ∆ is a sequentially egα-closed subset of the product space X × X.
Proof. Suppose that X is aegα-US space and {(xn, xn)} is a sequence in the diagonal ∆. It follows that {xn} is a sequence in X. Since X is egα-US, the sequence {(xn, xn)} egα-converges to (x, x) which clearly belongs to ∆. Therefore, ∆ is a sequentiallyegα-closed subset of X × X.
Conversely, suppose that the diagonal ∆ is a sequentiallyegα-closed sub- set of X × X. Assume that a sequence {xn} is egα-converging to x and y. Then it follows that {(xn, xn)} egα-converges to (x, y). By hypothe- sis, since ∆ is sequentially egα-closed, we have (x, y) ∈ ∆. Thus x = y.
Therefore, X is egα-US.
Theorem 2.8. If a space X is egα-U S and a subset M of X is se- quentially eGαO-compact, then M is sequentially egα-closed.
Proof. Assume that {xn} is any sequence in M which egα-converges to a point x∈ X. Since M is sequentially eGαO-compact, there exists a subsequence {xnk} of {xn} egα-converges to m ∈ M. Since X is egα-US, we have x = m. This shows that M is sequentially egα-closed.
Theorem 2.9. The product space of an arbitrary family of egα-U S topological space is aegα-US topological space.
Proof. Let {Xλ : λ ∈ ∆} be a family of egα-US topological spaces with the index set ∆. The product space of {Xλ : λ ∈ ∆} is denoted by ∏
Xλ. Let {xn(λ)} be a sequence in ∏
Xλ. Suppose that {xn(λ)} egα-converges to two distinct points x and y in ∏
Xλ. Then there exists a λ0 ∈ ∆ such that x(λ0) ̸= y(λ0). Then {xn(λ0)} is a sequence in Xλ0. Let Vλ0 be any egα-open in Xλ0 containing x(λ0). Then V = Vλ0 ×∏
λ̸=λ0Xλ is a egα-open set of ∏
Xλ containing x. Therefore, {xn(λ)} is eventually in V . Thus {xn(λ0)} is eventually in Vλ0 and it egα-converges to x(λ0). Similarly, the sequence {xn(λ0)} egα-converges to y(λ0). This is a contradiction as Xλ0 is a egα-US space. Therefore, the product space ∏
Xλ is egα-US.
3. Sequentially eGαO-compact preserving functions
Definition 3.1. A function f : X → Y is said to be
(i) Sequentiallyegα-continuous at x ∈ X if the sequence {f(xn)} egα- converges to f (x) whenever a sequence {xn} egα- converges to x.
If f is sequentially egα-continuous at each x ∈ X, then it is said to be sequentially egα-continuous.
(ii) Sequentially nearlyegα-continuous, if for each sequence {xn} in X that egα-converges to x ∈ X, there exists subsequence {xnk} of {xn} such that the sequence {f(xnk)} egα-converges to {f(xn)}.
(iii) Sequentially sub egα-continuous if for each point x ∈ X and each sequence{xn} in X egα-converging to x, there exists a subsequence {xnk} of {xn} and a point y ∈ Y such that the sequence {f(xnk)}
egα-converges to y.
(iv) Sequentially eGαO-compact preserving if the image f (M ) of every sequentially eGαO-compact set M of X is a sequentially eGαO- compact subset of Y .
Theorem 3.2. Let f1 : X → Y and f2: X → Y be two sequentially egα-continuous functions. If Y is egα-US, then the set E = {x ∈ X : f1(x) = f2(x)} is sequentially egα-closed.
Proof. Suppose that Y isegα-US and {xn} is any sequence in E that f1-converges to x ∈ X. Since f1 and f2 are sequentially egα-continuous functions, the sequence {f1(xn)} (respectively, {f2(xn)}) converges to f1(x) (respectively, f2(x)). Since xn ∈ E for each n ∈ N and Y is egα- U S, f1(x) = f2(x) and hence x∈ E. This shows that E is sequentially egα-closed.
Lemma 3.3. Every function f : X → Y is sequentially sub egα- continuous if Y is sequentially eGαO-compact.
Proof. Let {xn} be a sequence in X that egα-converges to x ∈ X.
It follows that {f(xn)} is a sequence in Y . Since Y is sequentially GαO-compact, there exists a subsequencee {f(xnk)} of {f(xn)} that egα- converges to a point y ∈ Y . Therefore f : X → Y is sequentially sub egα-continuous.
Theorem 3.4. Every sequentially nearly egα-continuous function is sequentially eGαO-compact preserving.
Proof. Let f : X → Y be a sequentially nearly egα-continuous func- tion and M be any sequentially eGαO-compact subset of X. We will show that f (M ) is a sequentially eGαO-compact subset of Y . So, assume that {yn} is any sequence in f(M). Then for each n ∈ N, there exists a point xn∈ M such that f(xn) = yn. Now M is sequentially eGαO-compact, so
there exists a subsequence {xnk} of {xn} that egα-converges to a point x∈ M. Since f is sequentially nearly egα-continuous, there exist a sub- sequence {xnk(i)} of {xnk} such that {f(xnk(i))} egα-converges to f(x).
Therefore, there exists a subsequence{ynk(i)} of {yn} that egα-converges to f (x). This implies that f (M ) is a sequentially eGαO-compact set of Y .
Theorem 3.5. Every sequentially eGαO-compact preserving function is sequentially sub egα-continuous.
Proof. Suppose that f : X → Y is a sequentially eGαO-compact pre- serving function. Let x be any point of X and {xn} a sequence that egα-converges to x. We denote the set {xn : n ∈ N} by A and put M = A∪ {x}. Since {xn} egα-converges to x, M is sequentially eGαO- compact. By hypothesis, f is sequentially eGαO-compact subset of Y . Now in f (M ) there exists a subsequence {f(xnk)} of {f(xn)} that egα- converges to a point y ∈ f(M). This implies that f sequentially sub egα-continuous.
Theorem 3.6. A function f : X → Y is sequentially eGαO-compact preserving if and only if f /M : M → f(M) is sequentially sub egα- continuous for each sequentially eGαO-compact set M of X.
Proof. Necessity: Suppose that f : X → Y is a sequentially eGαO- compact preserving function. Then f (M ) is sequentially eGαO-compact in Y for each sequentially eGαO-compact subset M of X. Therefore, by Theorem 3.5 f /M : M → f(M) is sequentially sub egα-continuous.
Sufficiency: Let M be any sequentially eGαO-compact set of X. We will show that f (M ) is sequentially eGαO-compact subset of Y . Let{yn} be any sequence in f (M ). Then for each n∈ N, there exists a point xn∈ M such that f (xn) = yn. Since {xn} is a sequence in the sequentially GαO-compact set M there exists a subsequencee {xnk} of {xn} that egα-converges to a point in M. By hypothesis f/M : M → f(M) is sequentially sub egα-continuous, hence there exists a subsequence {ynk} of {yn} that egα-converges to y ∈ f(M). This implies that f(M) is sequentially eGαO-compact in Y .
Corollary 3.7. If a function f : X → Y is sequentially sub egα- continuous and f (M ) is sequentially egα-closed in Y for each sequen- tially eGαO-compact set M of X, then f is sequentially eGαO-compact preserving.
Proof. It will sufficient to show that f /M : M → f(M) is sequentially sub egα-continuous for each sequentially eGαO-compact set M of X and by Lemma 3.3. we are done. So, let {xn} be any sequence in M that egα-converges to a point x ∈ M. Then, since f is sequentially sub egα- continuous there exists a subsequence {xnk} of {xn} and a point y ∈ Y such that {f(xnk)} egα-converges to y. Since {f(xnk)} is a sequence in the sequentially egα-closed set f(M) of Y , we obtain y ∈ f(M). This implies that f /M : M → f(M) is sequentially sub egα-continuous.
4. Slightly egα-Continuous Functions
Definition 4.1. A function f : X → Y is said to be slightly egα- continuous if for each x∈ X and for each v ∈ CO(Y, f(x)), there exists U ∈ eGαO(X, x) such that f (U )⊂ V ,where CO(Y, f(x)) is the family of clopen sets containing f (x) in a spaceY .
Definition 4.2. Let (D, ≤) be a directed set A net {xλ : λ∈ D} in X is said to beegα-convergent to a point x ∈ X if {xλ}λ∈D is eventually in each V ∈ egαO(X, x).
Theorem 4.3. For a function f : X → Y , the following are equiva- lent:
(a) f is slightly egα-continuous.
(b) f−1(v)∈ egαO(X) for each V ∈ CO(Y ).
(c) f−1(v) isegα-clopen for each V ∈ CO(Y ).
(d) for each x∈ X and for each net {xλ}λ∈D in X.
Proof. (a) ⇒ (b). Let V ∈ CO(Y ) and let x ∈ f−1(V ). Then f (x)∈ V . Since f is slightly egα-continuous, there is a U ∈ egαO(X, x) such that f (U ) ⊂ V . Thus f−1(U ) = ∪x{U : x ∈ f−1(V )}, that is f−1(U ) is a union ofegα-open sets. Hence f−1(U )∈ egαO(X).
(b) ⇒ (c). Let V ∈ CO(Y ). Then (Y − V ) ∈ CO(X). By hypothesis f−1(Y − V ) = X − f−1(V )∈ egαO(X). Thus f−1(V ) is egα-closed.
(c) ⇒ (d). Let {xλ}λ∈D be a net in X egα- converging to x and let V ∈ CO(Y, f(x)). There is thus a U ∈ egαO(X, x) such that f(U) ⊂ V . There is thus a λ0 ∈ D such that λ0 ≤ λ implies xλ∈ U since {xλ}λ∈D is egα- convergent to x. Thus f(xλ)∈ f(U) ⊂ V for all λ. Thus {f(xλ)}λ∈D
is egα- convergent to f(x).
(d) ⇒ (a). Suppose that f is not slightly egα-continuous at a point x ∈ X, then there exists a V ∈ CO(Y, f(x)) such that f(U) does not contained in V for each U ∈ egαO(X, x). So f(U)∩(Y −V ) ̸= ϕ and thus
U∩f−1(Y−V ) ̸= ϕ for each U ∈ egαO(X, x), since egαO(X, x) is directed by set inclusion C, there exists a selection function xU from egαO(X, x) into X for each U ∈ egαO(X, x). Thus {xU}U ∈ egαO(X, x) is a net in X egα-converging to x. Since XU ∈ U ∩ f−1(Y − V ) = U − f−1(V ) and so f (xU) /∈ V , for each U, {f(xU)}U ∈ egαO(X, x) is not eventually in V ∈ CO(Y, f(x)), which is a contradiction. Hence (a) holds.
Theorem 4.4. If f : X → Y is slightly egα-continuous and g : Y → Z is slightly continuous, then their composition g ◦ f is slightly egα- continuous.
Proof. Let V ∈ CO(Z), then g−1(V )∈ CO(Y ) [6]. Since f is slightly egα-continuous, f−1(g−1(V )) = (g◦ f)−1(V ) ∈ egαO(X). Thus g ◦ f is slightly egα-continuous.
Theorem 4.5. The following are equivalent for a function f : X → Y : (a) f is slightly egα-continuous,
(b) for each x ∈ X and for each V ∈ CO(Y, f(x)), there exists egα- clopen set U such that f (U )⊂ U,
(c) for each closed set F of Y , f−1(F ) is egα-closed, (d) f (cl(A))⊂ egαcl(f(A)) for each A ⊂ X and
(e) cl(f−1(B))⊂ f−1(egαcl(B)) for each B ⊂ Y .
Proof. (a) ⇒ (b) Let x ∈ X and V ∈ CO(Y, f(x)) by Theorem 4.3.
f−1(V ) is clopen. Put U = f−1(V ), then x∈ U and f(U) ⊂ V . (b) ⇒ (c) It is obvious.
(c) ⇒ (d) Since egαcl(f(A)) is the smallest egα-closed set containing f (A), hence by (c), we have (d).
(d) ⇒ (e) For each B ⊂ Y , f(cl(f−1(B))) ⊂ egαcl(f(f−1(B))) ⊂ egαcl(B). Hence f(cl(f−1(B)))⊂ egαcl(B) ⇒ cl(f−1(B))⊂ f−1(egαcl(B)).
(e) ⇒ (a) Let V ∈ CO(Y ). Then (Y − V ) ∈ CO(X), by (e), we have cl(f−1(Y − V )) ⊂ f−1(egαcl(Y − V )) = f−1(Y − V ), since every closed set is egα-closed, thus f−1(Y − V ) = X − f−1(V ) is closed and thus egα-closed, thus f−1(V )∈ egαO(X) and f is slightly egα-continuous.
Theorem 4.6. If f : X → Y is a slightly egα-continuous injection and Y is clopen T1, then X isegα-T1.
Proof. Suppose that Y is clopen T1. For any distinct points x and y in X, there exist V , W ∈ CO(Y ) such that f(x) ∈ V , f(y) /∈ V , f (x) /∈ W and f(y) ∈ W . Since f is slightly egα-continuous, f−1(V ) and f−1(W ) are egα-open subsets of X such that x ∈ f−1(V ), y /∈ f−1(V ), x /∈ f−1(W ) and y∈ f−1(W ). This shows that X isegα-T1.
Theorem 4.7. If f : X → Y is a slightly egα-continuous surjection and Y is clopen T2, then X isegα-T2.
Proof. For any pair of distinct points x and y in X, there exist disjoint clopen sets U and V in Y such that f (x)∈ U and f(y) ∈ V . Since f is slightlyegα-continuous, f−1(U ) and f−1(V ) areegα-open in X containing x and y respectively. Therefore f−1(U )∩f−1(V ) = ϕ because U∩V = ϕ.
This shows that X is egα -T2.
Definition 4.8. A space is called egα-regular if for each egα-closed set F and each point x /∈ F , there exist disjoint open sets U and V such that F ⊂ U and x ∈ V .
Definition 4.9. A space is said to be egα-normal if for every pair of disjointegα-closed subsets F1 and F2 of X, there exist disjoint open sets U and V such that F1 ⊂ U and F2⊂ V .
Theorem 4.10. If f is slightly egα-continuous injective open function from a egα-regular space X onto a space Y , then Y is clopen regular.
Proof. Let F be clopen set in Y and be y /∈ F . Take y = f(x). Since f is slightlyegα-continuous, f−1(F ) is aegα-closed set. Take G = f−1(F ), we have x /∈ G. Since X is egα-regular, there exist disjoint open sets U and V such that G⊂ U and x ∈ V . We obtain that F = f(G) ⊂ f(U) and y = f (x) ∈ f(V ) such that f(U) and f(V ) are disjoint open sets.
This shows that Y is clopen regular.
Theorem 4.11. If f is slightly egα-continuous injective open function from a egα-normal space X onto a space Y , then Y is clopen normal.
Proof. Let F1 and F2 be disjoint clopen subsets of Y . Since f is slightly egα-continuous, f−1(F1) and f−1(F2) are egα-closed sets. Take U = f−1(F1) and V = f−1(F2). We have U ∩ V = ϕ. Since X is egα- regular, there exist disjoint open sets A and B such that U ⊂ A and V ⊂ B. We obtain that F1 = f (U )⊂ f(A) and F2 = f (V )⊂ f(B) such that f (A) and f (B) are disjoint open sets. Thus, Y is clopen normal.
References
[1] M. Caldas and S. Jafari, On g-U S spaces, Universitatea Din Bacau Stud II SI Cercetari Stiintifice (Mathematica), No. 14 (2004), 13-20.
[2] R. Devi and A. Selvakumar, Onegα-closed sets in Topological spaces (Submit- ted).
[3] Erdal Ekici and Miguel Caldas, Slightly γ-Continuous Functions, Bol. Soc.
Oaran. Mat., (35) 22 (2004), no. 2, 63-74.
[4] R. C. Jain, The role of regularly open sets in general topology, Ph.D Thesis, Meerut University, Institute of Advanced Studies, Meerut, India, 1980.
[5] O. Njastad, On some classes of nearly open sets, Pacif. J. Math. 15 (1965), 961-970.
[6] T. Noiri, On semi-continuous mappings, Atti Accad NAz. Lincei Rend. Cl. Sci.
Fis. Mat. Natur, (8) 54 (1973), 210-214.
[7] A. Wilansky, Between T1 and T2, Amer. Math. Monthly Vol. (1967), 261-266.
*
Department of Mathematics Myongji University
Kyunggi 449-728, Republic of Korea E-mail : [email protected]
**
Department of Mathematics Kongunadu and Science College
Coimabtore 641029, Tamil Nadu, India E-mail : [email protected]
***
Department of Mathematics Kongunadu and Science College
Coimabtore 641029, Tamil Nadu, India
