A NOTE ON SEMITOPOLOGICAL PROPERTIES. D. Sivaraj

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1 A NOTE ON SEMITOPOLOGICAL PROPERTIES D. Sivaraj (received 11 May 1982, revised 16 November 1982) 1. Introduction Let (*,T) be a topological space and A a subset of X. The closure and interior of A in (.,t) are denoted by C(A) and 1(A) respectively. A is said to be regular open if A = l(c(a)) and regular closed if A = C(KA)). RO{X,t) is the family of all regular open sets in (*,t). A subset A of X in a space (^,t) is said to be semi-open, [7], if there exists G t such that G c A <- C(G). The family of all semi-open sets in (X,t) is denoted by SO(X,t). A subset A of X in a space (^,t) is said to be an a-set, [8], if A A* where A* - l(c(_i(a))). If xa is the family of all a-sets in (X,t), then t c t c 50(^,t), t is a topology on X, and SO{Xt t) = SO(X, ta). A topological space (;,t) is S-closed, [16], if every semi-open cover of X has a finite subfamily whose closures cover X. A subset A of X is an S-closed subspace of (X,t), if (i4,t i4) is S-closed where t i4 is the relative topology on A. A is S-closed relative to t, [9], if every cover of A by semi-open sets of X has a finite subfamily whose closures cover A. A space (*,t) is locally S-closed, [10], if each x i X has an open neighbourhood which is an S-closed subspace of X. A space (^T,t) is quasi-h-closed (QHC), [ll], (feebly compact [l]) if every open (countable open) cover of X has a finite subfamily whose closures cover X. A space (,t) is nearly compact, [l3] (mildly compact, [l4]) if every regular open (countable regular open) cover of X has a finite subcover. A space (X,t) is weakly Hausdorff, [15], if each point of X is an intersection of regular closed sets of (*,t). A space (X,t ) is extremally disconnected if the closure of every open set is open. A space (X,t) is almost regular, [12 ] if for each point x X and each regular closed set A not containing x, there exist disjoint open sets U and V such that x U and A <- V. A space (X,t) is Math. Chronicle 13(1984)

2 pseudocompact, [18], X is bounded. if and only if every continuous real-valued function on If X is a set of points, let T(X) be the lattice of topologies on X. If t T(X) and [t] is the equivalence class of topologies on X which yield the same semi-open sets as (I,t), then there is a finest element of [t], denoted by F(t) [4]. A property of topological spaces is defined to be a semi-topological property, [5], if it is preserved by semi-homeomorphisms (bijections so that the images of semi-open sets are semi-open and the inverse images of semi-open sets are semi-open). The following lemmas will be used in the sequel. Lemma 1.1 [5]. If f: (X,x) * (Y,o) is a semi-homeomorphism3 then f : (Xy F(t) ) (Y, F(a) ) is a homeomorphim. Lemma 1.2 [10]. In a space (X,\), if A is S-olosed relative to t, then I(C(A)) is also S-olosed relative to t. Lemma 1.3 [10]. A space (X3t) is locally S-closed if and only if for each x X, there exists V t such that x V and V is S-closed relative to x. Lemma 1.4 [lo]. If X is a locally S-closed space and f :X~*-Y is an open and continuous surjection, then Y is locally S-closed. 2. The Results The following first two lemmas play the key role in this paper. Lemma 2.1. For a space (X,t)3 t = F(t). Proof. The result follows from proposition 4 of [8] and theorem 2 of [2]. Lemma 2.2. In a space (X,t), if A SO(X3t ), then C*(A) = C(A) where C*(A) is the closure of A with respect to the topology F(i). 74

3 Proof. Since x c F(t), for any A c x, C*{A) c C(A). For the conyerse, if x t C*(A), there exists V F ( t ) such that x V and A fl V» which implies that 104) fl I(K) = which implies that I(j4) fl C(I(V)) = <p and so 1(A) fl V* - <f>, which implies that C(J(i4)) fl 7* = <j>. Since 4 is semi-open, A c 7(J(j4)) and so, A fl V* = <p. Since V* is a t-open neighbourhood of x, x t C(A). Therefore C(A) c *(>4), which proves the lemma. Lemma 2.3. A space (X,x) is QHC (feebly compact) if and only if (XsF(t)) is QHC (feebly compact). Proof. If (X.FCt)) is QHC, by lemma 2.2, (Jf,x) is QHC. Conversely, if (X,x) is QHC, since for each V t F(x), V c V*, V* t and (7(7) = C(V*), [X,F(t)) is QHC by lemma 2.2. The proof for feebly compact spaces is similar. Theorem 2.1. The property of a space being QHC (feebly compact) is a semi-topological property. Proof. Since the homeomorphic image of a QHC (feebly compact) space is a QHC (feebly compact) space, proof follows from lemmas 2.3, and 1.1. following Since RO(X,t) = ^ ( ^ ^ ( t ) ), by proposition 6 of [8], we have the Lemma 2.4. A space (X3x) is nearly compact (mildly compacts weakly Hausdorff) if and only if (X3F(x)) is nearly compact (mildly compact, weakly Hausdorff). Theorem 2.2. The property of a space being nearly compact (mildly compact> weakly Hausdorff) is a semi-topological property. Proof. Since the homeomorphic image of a nearly compact (mildly compact, weakly Hausdorff) space is a nearly compact (mildly compact, weakly Hausdorff) space, proof follows from lemmas 2.4, and

4 Theorem 2.3. The property of a space being extremally disconnected is a semi-topological property. Proof. By proposition 7 of 8], (*,x) is extremally disconnected if and only if (Z,F(x)) is extremally disconnected. Since the homeomorphic image of an extremally disconnected space is an extremally disconnected space, the proof follows from lemma 1.1. Lemma 2.5. A space (X3t) is almost regular if and only if (X,F(t)) almost regular. is Proof. If (X,x) is almost regular, there is nothing to prove. Conversely, suppose (Z,F(t)) is almost regular. If A is regular closed and x f. A, there exist disjoint sets U and V F(x) such that x U and A c v. Then, U* and V* are the required disjoint open sets such that x U* and A c V*. Hence (X,x) is almost regular. Theorem 2.4. The property of a space being almost regular is a semitopological property. Proof. Since the homeomorphic image of an almost regular space is an almost regular space, the proof follows from lemmas 2.5, and 1.1. Lemma 2.6. In a space (Xjt), a subset A of X is S-closed relative to x if and only if A is S-closed relative to F(i). Proof. Since SO(X,x) = 50(*,2: (x)), the proof follows from lemma 2.2. Lemma 2.7. A space (X3t) is locally S-closed if and only if (XyF(r)) locally S-closed. is Proof. Suppose (Jf,f(T)) is locally 5-closed. Let x X. Then, by lemma 1.3, there exists V F(x) such that x i V and V is 5-closed relative to F(x). By lemma 2.6, V is 5-closed relative to x. By lemma 1.2, J(C(7)) is 5-closed relative to x. Since x V c /(CCF)), (*,x) is locally 5-closed. Conversely, if (X,x) is locally 5-closed, 76

5 since t c F(t), by lemma 2.6, (^,F(t)) is locally S-closed. Theorem 2.5. topological property. The property of a space being locally S-closed is a semi- Proof. The proof follows from lemmas 2.7, and 1.4. Theorem 2.6. topological property. The property of a space being pseudocompact is a semi- Proof. If C(X3t) is the set of all continuous real-valued functions on (Z,t), since the real line with the usual topology is regular, by proposition 8 of [8],C{X, t) = C(XtF(t)) and so, is pseudocompact if and only if (,F(t)) is pseudocompact. Then, since the homeomorphic image of a pseudocompact space is a pseudocompact space, the proof follows from lemma 1.1. REFERENCES 1. R.W. Bagley, E.H. Cannel and J.D. Mcknight, Jr, On properties characterizing pseudocompact spaces, Proc. Amer. Math. Soc. 9(1958) S.G. Crossley, A note on semi-topological classes, Proc. Amer. Math. Soc. 43(1974), S.G. Crossley, A note on semi-topological properties, Proc. Amer. Math. Soc. 72(1978), S.G. Crossley and S.K. Hildebrand, Semi-closure, Texas J. Sci. 22(1971), S.G. Crossley and S.K. Hildebrand, Semi-topological properties, Fund. Math. 74(1972), T.R. Hamlett, The property of being a Baire space is semi-topological, Math. Chronicle, 5(1977),

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