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Almost p-normal, mildly p-normal spaces and some functions
Chaos, Solitons and Fractals 18 (2003) 267–274 www.elsevier.com/locate/chaos
Almost p-normal, mildly p-normal spaces and some functions Jin Han Park Division of Mathematical Sciences, Pukyong National University, 599-1 Daeyeon 3-Dong Nam-Gu, Pusan 608-737, South Korea Accepted 14 January 2003
Abstract The aim of this paper is to obtain some characterizations of almost p-normal spaces and mildly p-normal spaces and to improve the preservation theorems of p-normal spaces and mildly p-normal spaces established by Navalagi [P normal, almost p-normal and mildly p-normal spaces. Topology Atlas Preprint #427. URL: http://at.yorku.ca/i/d/e/b/ 71.htm.]. Ó 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction and preliminaries In 1973, Singal and Singal [27] introduced a weak form of normal spaces called mildly normal spaces. In 1989, Nour [20] used preopen sets to define p-normal spaces. Recently, Navalagi [16] have continued the study of further properties of p-normal spaces and also defined and investigated mildly p-normal (resp. almost p-normal) spaces which are generalizations of both mildly normal (resp. almost normal) spaces and p-normal spaces. On the other hand, Levine [10] initiated the investigation of so-called g-closed sets in topological spaces, since then many modifications of g-closed sets were defined and investigated by a large number of topologists (see, References of this paper). In [11,19], Maki and coworkers defined and investigated the concept of gp-closed sets and used this notion to obtain a characterization of pnormal spaces. This notion is generalization of preclosed sets which were further studied by Dontchev and Maki [8], Arokiarani et al. [1], Noiri et al. [19] and Park et al. [22]. Recently, Gnanambal [9] have defined and studied the notion of gpr-closed sets which are implied by both that of rg-closed sets and gp-closed sets. Noiri [18] called gpr-closed set rgpclosed and used the sets to obtain further characterizations of almost p-regular spaces due to Malghan and Navalagi [12]. In this paper, we further study properties of gpr-closed sets and gpr-open sets and obtain new characterizations of almost p-normal spaces and mildly p-normal spaces by using gpr-open sets. We use pre-rgp-closed functions [18] to obtain certain preservation theorems of p-normal spaces and mildly p-normal spaces. The main result of this paper is that p-normality (resp. mild p-normality) is preserved under pre-gp-closed continuous (resp. pre-rgp-closed R-map) surjection. Throughout this paper, spaces ðX ; sÞ and ðY ; rÞ (or simply X and Y ) always mean topological spaces on which no separation axioms are assumed unless explicitly stated. Let A be a subset of a space X . The closure of A and the interior of A are denoted by clðAÞ and intðAÞ, respectively. A subset A is said to be regular open (resp. regular closed) if A ¼ intðclðAÞÞ (resp. A ¼ clðintðAÞÞ). A subset A is said to be preopen [13] if A intðclðAÞÞ. The complement of a preopen set is called preclosed. The intersection of all preclosed sets containing A is called the preclosure [13] of A and is denoted by pclðAÞ. Dually, the preinterior of A, denoted by pintðAÞ is defined to be the union of all preopen sets contained in A. A subset A is said to be a preneighborhood [15] of x if there exists a preopen set U such that x 2 U A.
E-mail address: [email protected] (J.H. Park). 0960-0779/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0960-0779(02)00650-1
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We recall the following definition used in sequel. Definition 1.1. A subset A of a space X is called: (a) generalized preclosed [11] (briefly, gp-closed) if pclðAÞ U whenever A U and U is open in X , (b) regular generalized closed [21] (briefly rg-closed) if clðAÞ U whenever A U and U is regular open in X , (c) generalized preregular closed [9] (briefly, gpr-closed), or regular generalized preclosed [18] if pclðAÞ U whenever A U and U is regular open in X , (e) gp-open (resp. rg-open, gpr-open) if the complement of A is gp-closed (resp. rg-closed, gpr-closed). 2. Generalized preregular closed sets We first recall following lemmas to obtain further results for gpr-closed sets. Lemma 2.1 [12]. If Y is an open subspace of a space X and A is a subset of Y , then pclY ðAÞ ¼ pclðAÞ \ ðY Þ. Lemma 2.2 [9]. If A is a regular open and gpr-closed subset of a space X , then A is preclosed in X . Theorem 2.3. Let Y be an open subspace of a space X and A Y . If A is gpr-closed in X , then A is gpr-closed in Y . Proof. Let U be a regular open set of Y such that A U . Then U ¼ V \ Y for some regular open set V of X . Since A is gpr-closed in X , we have pclðAÞ V and by Lemma 2.1, pclY ðAÞ ¼ pclðAÞ \ Y V \ Y ¼ U . Hence A is gpr-closed in Y. Theorem 2.4. Let Y be a gpr-closed and regular open subspace of a space X . If A is gpr-closed in Y , then A is gpr-closed in X . Proof. Let U be any regular open subset of X such that A U . Since U \ Y is regular open in Y and A is gpr-closed in Y , pclY ðAÞ U \ Y . By Lemmas 2.1 and 2.2, we have pclðAÞ ¼ pclðAÞ \ Y ¼ pclY ðAÞ U \ Y U. Hence A is gpr-closed in X . Corollary 2.5. If A is a gpr-closed regular open set and B is a preclosed set of a space X , then A \ B is gpr-closed. Theorem 2.6. Let A be a gpr-closed set. Then A ¼ pclðpintðAÞÞ if and only if pclðpintðAÞÞ n A is regular closed. Proof. If A ¼ pclðpintðAÞÞ, then pclðpintðAÞÞ n A ¼ / and hence pclðpintðAÞÞ n A is regular closed. Conversely, let pclðpintðAÞÞ n A be regular closed. Since pclðAÞ n A contains the regular closed set pclðpintðAÞÞ n A. By Theorem 3.15 in [9], pclðpintðAÞÞ n A ¼ / and hence pclðpintðAÞÞ ¼ A. A space X is said to be almost p-regular [12] if for each regular closed set A of X and each point x 2 X n A, there exist disjoint preopen sets U and V such that x 2 U and A V . A subset A of a space X is said to be p-closed relative to X [7] if for every cover fVa : a 2 Kg of A by preopen subsets of X , there exists a finite subset K0 of K such that A [fpclðVa Þ : a 2 K0 g. Theorem 2.7. If a space X is almost p-regular and a subset A of X is p-closed relative to X , then A is gpr-closed. Proof. Let U be any regular open set of X containing A. For each x 2 A, there exists a preopen set V ðxÞ such that x 2 V ðxÞ pclðV ðxÞÞ U . Since fV ðxÞ : x 2 Ag is a preopen cover of A, there exists a finite subset A0 of A such that A [fpclðV ðxÞÞ : x 2 A0 g. Hence we obtain A pclðAÞ [fpclðV ðxÞÞ : x 2 A0 g U . This shows that A is gpr-closed. Recall that a space X is called regular T1=2 [21] if every rg-closed set of X is regular closed. A set A of a space X is called d-closed [28] if A ¼ cld ðAÞ, where cld ðAÞ ¼ fx 2 X : intðclðU ÞÞ \ A 6¼ /; U is open set containing xg. Theorem 2.8. The following are equivalent for a space X in which every gpr-closed set is semiopen: (a) X is regular T1=2 . (b) Every singleton is regular closed or regular open in X .
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(c) Every singleton is regular closed or d-open in X . (d) Every gpr-closed subset of X is regular closed. Proof. (a) ) (b): Let x 2 X . If fxg is not regular closed, then A ¼ X n fxg is not regular open and hence A is trivially rgclosed. By the regular T1=2 property of X , A is regular closed and hence fxg is regular open. (b) () (c) follows from Lemma 4.2 in [6]. (b) ) (d): Let A X be gpr-closed. Let x 2 clðintððAÞÞ. We consider the following two cases: Case I. Let fxg be regular open. Since x 2 clðintððAÞÞ, then fxg \ intðAÞ 6¼ /. Thus x 2 A. Case II. Let fxg be regular closed. If we assume that x 62 A, then we have x 2 clðintðAÞÞ n A ¼ pclðAÞ n A (because pclðAÞ ¼ A [ clðintðAÞÞ), which is impossible according to Theorem 3.15 in [9]. Hence x 2 A. i.e., A is preclosed. Since A is gpr-closed, by hypothesis, A is semiopen and thus regular closed. (d) ) (a): Since every rg-closed is gpr-closed, the proof is clear. Now, we start with following lemma to get some results for gpr-open sets. Lemma 2.9 [9,16]. A subset A of a space X is gp-open (resp. gpr-open) if and only if F pintðAÞ whenever F is closed (resp. regular closed) and F A. Recall that two nonempty sets A and B of X are said to be preseparated [23] if pclðAÞ \ B ¼ / ¼ A \ pclðBÞ. Theorem 2.10. If A and B are gpr-open preseparated sets, then A [ B is gpr-open. Proof. Let F be a regular closed subset of A [ B. Then F \ pclðAÞ A and hence by Lemma 2.9, F \ pclðAÞ pintðAÞ. Similarly, F \ pclðBÞ pintðBÞ. Then we have F pintðA [ BÞ and hence A [ B is gpr-open. Theorem 2.11. If A is a gpr-open set of a space X , then U ¼ X whenever U is regular open and pintðAÞ [ ðX n AÞ U . Proof. Let U be a regular open set and pintðAÞ [ ðX n AÞ U. Then X n U ðX n pintðAÞ \ AÞ, i.e., ðX n U Þ pclðX n AÞ \ ðX n AÞ. Since X n A is gpr-closed, by Theorem 3.15 in [9], X n U ¼ /. Hence U ¼ X . The converse in the theorem above is not always true as shown by the following example. Example 2.12. Let X ¼ fa; b; c; dg and s ¼ fX ; /; fag; fbg; fa; bg; fa; b; cg; fa; b; dgg. Let A ¼ fb; c; dg. Then X is the only regular open set containing pintðAÞ [ ðX n AÞ but A is not gpr-closed in X . Theorem 2.13. Let Y be a regular open and gpr-closed subspace of a space X . If A is gpr-open in Y , then A is gpr-open in X . Proof. Let F be any regular closed set and F A. Since F is regular closed in Y and A is gpr-open in Y , F pintY ðAÞ and then F pintðAÞ \ Y . Hence F pintðAÞ and so A is gpr-open in X . 3. Characterizations of almost p-normal and mildly p-normal spaces Definition 3.1. A space X is said to be (a) p-normal [16], or prenormal [20] (resp. mildly p-normal [16]) if for every pair of disjoint closed (resp. regular closed) sets A and B of X , there exist disjoint preopen sets U and V such that A U and B V . (b) almost p-normal [16] if for each closed set A and regular closed set B of X such that A \ B ¼ /, there exist disjoint preopen sets U and V such that A U and B V . Theorem 3.2. The following are equivalent for a space X : (a) X is almost p-normal. (b) For each closed set A and regular closed set B such that A \ B ¼ /, there exist disjoint gp-open sets U and V such that A U and B V . (c) For each closed set A and regular closed set B such that A \ B ¼ /, there exist disjoint gpr-open sets U and V such that A U and B V .
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(d) For each closed set A and each regular open set B containing A, there exists a gpr-open set V A V pclðV Þ B. (e) For each rg-closed set A and each regular open set B containing A, there exists a preopen set V clðAÞ V pclðV Þ B. (f) For each rg-closed set A and each regular open set B containing A, there exists a gp-open set V clðAÞ V pclðV Þ B. (g) For each rg-closed set A and each regular open set B containing A, there exists a gpr-open set V clðAÞ V pclðV Þ B.
of X such that of X such that of X such that of X such that
Proof. It is obvious that (a) ) (b) ) (c) and (e) ) (f) ) (g) ) (d). (c) ) (d): Let A be a closed set and B be a regular open subset of X containing A. There exist disjoint gpr-open sets V and W such that A V and X n B W . By Lemma 2.9, we have X n B pintðW Þ and V \ pintðW Þ ¼ /. Therefore, we obtain pclðV Þ \ pintðW Þ ¼ / and hence A V pclðV Þ X n pintðW Þ B. (d) ) (a): Let A be any closed set and B be any regular closed set such that A \ B ¼ /. Then X n B is a regular open set containing A and there exists a gpr-open set G of X such that A G pclðGÞ X n B. Put U ¼ pintðGÞ and V ¼ X n pclðGÞ. Then U and V are disjoint preopen sets of X such that A U and B V . Hence X is almost p-normal. (a) ) (e): Let A be a rg-closed set and B be a regular open subset of X containing A. Then clðAÞ B and by Theorem 4.3 in [16], there exists a preopen set V such that clðAÞ V pclðV Þ B. Remark 3.3. We can obtain characterizations of almost p-normal spaces by replacing ‘‘A is rg-closed’’ in (e)–(g) of Theorem 3.2 with ‘‘A is g-closed’’. The proof is similar to that of Theorem 3.2. Theorem 3.4. The following are equivalent for a space X : (a) (b) (c) (d)
X is mildly p-normal. For any disjoint regular closed sets A and B of X , there exist disjoint gp-open sets U and V such that A U and B V . For any disjoint regular closed sets A and B of X , there exist disjoint gpr-open sets U and V such that A U and B V . For each regular closed set A and each regular open set B containing A, there exists a gp-open set V of X such that A V pclðV Þ B. (e) For each regular closed set A and each regular open set B containing A, there exists a gpr-open set V of X such that A V pclðV Þ B.
Proof. The proof is similar to that of Theorem 3.2.
4. Pre-gpr-continuous and gpr-irresolute functions Definition 4.1. A function f : X ! Y is called: (a) precontinuous [13] (resp. preirresolute [24]) if f 1 ðF Þ is preclosed in X for every closed (resp. preclosed) set F of Y ; (b) gp-continuous [1] (resp. pre-gp-continuous [22], gp-irresolute [1]) if f 1 ðF Þ is gp-closed in X for every closed (resp. preclosed, gp-closed) set F of Y ; (c) gpr-continuous [9] (resp. pre-gpr-continuous, gpr-irresolute [9]) if f 1 ðF Þ is gpr-closed) in X for every closed (resp. preclosed, gpr-closed) set F of Y . From the definition stated above, we obtain the following diagram of implications: gp-irresoluteness gpr-irresoluteness + + preirresoluteness ) pre-gp-continuity ) pre-gpr-continuity + + + precontinuity ) gp-continuity ) gpr-continuity Remark 4.2 (a) None of the implications in above diagram are reversible; (b) Preirresoluteness, gp-irresoluteness and gpr-irresoluteness are mutually independent; (c) Pre-gpr-continuity and gp-continuity are independent of each other.
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Example 4.3 (a) Let X ¼ fa; b; c; dg, s ¼ fX ; /; fag; fbg; fa; bgg and r ¼ fX ; /; fc; dgg. Let f : ðX ; sÞ ! ðX ; rÞ be an identity. Then f is gpr-continuous but not pre-gpr-continuous since fag is preclosed in ðX ; rÞ and f 1 ðfagÞ is not gpr-closed in ðX ; sÞ. (b) Let ðX ; sÞ and ðX ; rÞ be spaces given in (a). Let f : ðX ; sÞ ! ðX ; rÞ be a function defined by f ðaÞ ¼ f ðbÞ ¼ a, f ðcÞ ¼ b and f ðdÞ ¼ d. Then f is gpr-irresolute but not pre-gp-continuous since fag is preclosed in ðX ; rÞ and f 1 ðfagÞ is not gp-closed in ðX ; sÞ. (c) Let X ¼ fa; b; c; dg, s ¼ fX ; /; fag; fbg; fa; bgg and r ¼ fX ; /; fag; fa; bgg. Let f : ðX ; sÞ ! ðX ; rÞ be a function defined by f ðaÞ ¼ a, f ðbÞ ¼ f ðcÞ ¼ c and f ðdÞ ¼ d. Then f is gp-irresolute (and hence pre-gpr-continuous) but not gpr-irresolute since fag is gpr-closed in ðX ; rÞ and f 1 ðfagÞ is not gpr-closed in ðX ; sÞ. (d) Let X ¼ fa; b; c; dg, s ¼ fX ; /; fag; fa; bg; fc; dg; fa; c; dgg and r ¼ fX ; /; fbg; fa; cg; fa; b; cgg. Let f : ðX ; sÞ ! ðX ; rÞ be an identity. Then f is pre-gpr-continuous but not gp-continuous since fa; c; dg is closed in ðX ; rÞ and f 1 ðfa; c; dgÞ is not gp-closed in ðX ; sÞ. (e) Let ðX ; sÞ and ðX ; rÞ be spaces given in (d). Let f : ðX ; rÞ ! ðX ; sÞ be a function defined by f ðaÞ ¼ a, f ðbÞ ¼ c, f ðcÞ ¼ b and f ðdÞ ¼ d. Then f is gp-continuous but not pre-gpr-continuous since fcg is preclosed in ðX ; sÞ and f 1 ðfcgÞ is not gpr-closed in ðX ; rÞ. (f) Let X ¼ fa; b; c; dg, s ¼ fX ; /; fag; fb; cg; fa; b; cgg and r ¼ fX ; /; fag; fbg; fa; bg; fa; b; cg; fa; b; dgg. Let f : ðX ; sÞ ! ðX ; rÞ be an identity. Then f is preirresolute but not gpr-irresolute since fb; cg is gpr-closed in ðX ; rÞ and f 1 ðfb; cgÞ is not gpr-closed in ðX ; sÞ. Recall that a space X is called preregular T1=2 [9] if every gpr-closed set is preclosed. Theorem 4.4. If a function f : X ! Y is gpr-continuous (resp. pre-gpr-continuous) and X is preregular T1=2 , then f is precontinuous (resp. preirresolute). Proof. Let F be any closed (resp. preclosed) set of Y . Since f is gpr-continuous (resp. pre-gpr-continuous), f 1 ðF Þ is gprclosed in X and then f 1 ðF Þ is preclosed in X . Hence f is precontinuous (resp. preirresolute). A space X is called submaximal [4] if every dense subset of X is open. Note that the submaximal spaces are exactly the space where every preopen set is open. Theorem 4.5. If a function f : X ! Y is pre-gpr-continuous (resp. gpr-continuous) and Y is preregular T1=2 (resp. submaximal), then f is gpr-irresolute (resp. pre-gpr-continuous). Proof. Let F be any gpr-closed (resp. preclosed) subset of Y . Since Y is preregular T1=2 (resp. submaximal), then F is preclosed (resp. closed) in Y . Hence f 1 ðF Þ is rgp-closed in X . This show that f is gpr-irresolute (resp. pre-gpr-continuous). Theorem 4.6. Let f : X ! Y and g : Y ! Z be functions. (a) If f is gpr-irresolute and g is gpr-continuous (resp. pre-gpr-continuous), then the composition g f : X ! Z is gpr-continuous (resp. pre-gpr-continuous). (b) If f and g are pre-gpr-continuous and Y is preregular T1=2 , then the composition g f : X ! Z is pre-gpr-continuous. (c) If f and g are gpr-irresolute, then the composition g f : X ! Z is gpr-irresolute. The composition of two gpr-continuous functions need not be pre-gpr-continuous as seen by the following example. Example 4.7. Let X ¼ fa; b; c; dg, s ¼ fX ; /; fag; fb; cg; fa; b; cgg, r ¼ fX ; /; fag; fbg; fa; bgg and q ¼ fX ; /; fcdgg. Let f : ðX ; sÞ ! ðX ; rÞ be a function defined by f ðaÞ ¼ f ðdÞ ¼ d, f ðbÞ ¼ b and f ðcÞ ¼ a and g : ðX ; rÞ ! ðX ; qÞ be a function defined by gðaÞ ¼ gðbÞ ¼ a, gðdÞ ¼ c and gðdÞ ¼ d. Then f and g are pre-gpr-continuous but the composition g f is not pre-gpr-continuous since fag is preclosed in ðX ; qÞ and ðg f Þ1 ðfagÞ is not gpr-closed in ðX ; sÞ. 5. Preservation theorems Definition 5.1. A function f : X ! Y is called (a) almost continuous [26] (resp. R-map [5], completely continuous [2]) if f 1 ðV Þ is open (resp. regular open, regular open) in X for every regular open (resp. regular open, open) set V of Y ,
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(b) almost preirresolute [16] if for each x 2 X and preneighborhood V of f ðxÞ, pclðf 1 ðV ÞÞ is a preneighborhood of x, (c) pre-gp-closed [19] (resp. pre-rgp-closed [18], M-preclosed [14]) if f ðF Þ is gp-closed (resp. gpr-closed, preclosed) in Y for every preclosed set F of X , (d) rc-preserving [17] (resp. almost closed [26]) if f ðF Þ is regular closed (resp. closed) in Y for every regular closed set F of X , (e) preopen [13], or almost open [25] (resp. M-preopen [14], semiopen [3]) if f ðU Þ is preopen (resp. preopen, semiopen) in Y for every open (preopen, open) set U of X , (f) weakly open [25] if f ðU Þ intðf ðclðU ÞÞÞ for every open set U of X . Note that preopeness and weak openness are independent of each other [25]. Lemma 5.2 [19]. A surjection f : X ! Y is pre-gp-closed (resp. pre-rgp-closed) if and only if for each subset B of Y and each preopen set U of X containing f 1 ðBÞ there exists a gp-open (resp. gpr-open) set of V of Y such that B V and f 1 ðV Þ U . Lemma 5.3 [19]. If a function f : X ! Y is continuous preopen, then f is M-preopen. Lemma 5.4. If a function f : X ! Y is weakly open continuous, then f is M-preopen and R-map. Proof. Since every weakly open continuous function is R-map [18], we show that f is preopen. Let U be any open set of X . Since f is weakly open and continuous, f ðU Þ f ðIntðClðU ÞÞÞ Intðf ðClðIntðClðU ÞÞÞÞÞ Intðf ðClðU ÞÞÞ IntðClðf ðU ÞÞÞ: Then f ðU Þ is preopen in Y and thus f is preopen. Hence, by Lemma 5.3, f is M-preopen.
Lemma 5.5. If a function f : X ! Y is semiopen precontinuous, then f is preirresolute. Proof. Let V be a preopen set of Y . Then sclðV Þ ¼ intðclðV ÞÞ. Since f is semiopen, f 1 ðintðclðV ÞÞÞ ¼ f 1 ðsclðV ÞÞ clðf 1 ÞÞ. Since f is precontinuous, we have f 1 ðV Þ f 1 ðintðclðV ÞÞÞ intðclðf 1 ðintðclðV ÞÞÞÞÞ intðclðf 1 ðV ÞÞÞ: Thus, f 1 ðV Þ is preopen and hence f is preirresolute.
Theorem 5.6 (a) If f : X ! Y is a preopen continuous almost preirresolute surjection and X is p-normal, then Y is p-normal. (b) If f : X ! Y is an R-map preopen continuous almost preirresolute surjection and X is almost p-normal, then Y is almost p-normal. Proof (a) Let A be a closed subset of Y and B be an open set containing A. Then f 1 ðAÞ is closed and f 1 ðBÞ is an open set of X such that f 1 ðAÞ f 1 ðBÞ. By the p-normality of X , there exists a preopen set U of X such that f 1 ðAÞ U pclðU Þ f 1 ðBÞ and then f ðf 1 ðAÞÞ f ðU Þ f ðpclðU ÞÞ f ðf 1 ðBÞÞ. Since f is preopen continuous almost preirresolute surjection, by Lemma 5.3 we have f ðU Þ is preopen set of Y such that A f ðU Þ pclðf ðU ÞÞ B. Hence Y is p-normal. (b) The proof is entirely analogous to (a). Corollary 5.7 (a) [16, Theorem 3.9] If f : X ! Y is an M-preopen continuous almost preirresolute function from a p-normal space X onto a space Y , then Y is p-normal. (b) [16, Theorem 4.6] If f : X ! Y is continuous M-preopen R-map and almost preirresolute surjection from an almost pnormal space X onto a space Y , then Y is almost p-normal. Theorem 5.8 (a) If f : X ! Y is an weakly open continuous almost preirresolute surjection and X is p-normal, then Y is p-normal. (b) If f : X ! Y is a preopen semiopen continuous surjection and X is a p-normal space, then Y is p-normal.
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Proof (a) Let A be a closed subset of Y and B be an open set containing A. Then f 1 ðAÞ is closed and f 1 ðBÞ is an open set of X such that f 1 ðAÞ f 1 ðBÞ. By the p-normality of X , there exists a preopen set U of X such that f 1 ðAÞ U pclðU Þ f 1 ðBÞ and then f ðf 1 ðAÞÞ f ðU Þ f ðpclðU ÞÞ f ðf 1 ðBÞÞ. Since f is weakly open continuous almost preirresolute surjection, by Lemma 5.4 we have f ðU Þ is preopen set of Y such that A f ðU Þ pclðf ðUÞÞ B. Hence Y is p-normal. (b) The proof follows from Theorem 5.6(a) using Lemmas 5.3 and 5.5. Theorem 5.9 (a) If f : X ! Y (b) If f : X ! Y (c) If f : X ! Y (d) If f : X ! Y (e) If f : X ! Y mal.
is a pre-gp-closed continuous surjection and X is p-normal, then Y is p-normal. is an R-map pre-rgp-closed surjection and X is mildly p-normal, then Y is mildly p-normal. is a completely continuous pre-rgp-closed surjection and X is mildly p-normal, then Y is p-normal. is an almost continuous pre-rgp-closed surjection and X is p-normal, then Y is mildly p-normal. is a continuous weakly open pre-rgp-closed surjection and X is mildly p-normal, then Y is almost p-nor-
Proof (a) Let A and B be any disjoint closed sets of Y . Then f 1 ðAÞ and f 1 ðBÞ are disjoint closed sets of X . Since X is p-normal, there exist disjoint preopen sets U and V such that f 1 ðAÞ U and f 1 ðBÞ V . By Lemma 5.2, there exist gp-open sets G and H of Y such that A G, B H, f 1 ðGÞ U and f 1 ðH Þ V . Since U and V are disjoint, G and H are disjoint. By Lemma 2.9, we have A pintðGÞ, B pintðH Þ and pintðGÞ \ pintðHÞ ¼ /. This shows that Y is p-normal. The proofs of (b)–(e) are entirely analogous to (a). Corollary 5.10 (a) [16, Theorem 3.11] If f : X ! Y is an M-preclosed continuous function from a p-normal space onto a space Y , then Y is p-normal. (b) [16, Theorem 5.9] If f : X ! Y is an R-map M-preclosed function from a mildly p-normal space X onto a space Y , then Y is mildly p-normal. Theorem 5.11 (a) If f : X ! Y is a pre-gpr-continuous rc-preserving injection and Y is mildly p-normal, then X is mildly p-normal. (b) If f : X ! Y is a pre-gpr-continuous almost closed injection and Y is p-normal, then X is mildly p-normal. Proof (a) Let A and B be any disjoint regular closed sets of X . Since f is re-preserving injection, f ðAÞ and f ðBÞ are disjoint regular closed sets of Y . By mild p-normality of Y , there exist disjoint preopen sets U and V of Y such that f ðAÞ U and f ðBÞ V . Since f pre-gpr-continuous, f 1 ðU Þ and f 1 ðV Þ are disjoint gpr-open sets containing A and B, respectively. Hence by Theorem 3.4, X is mildly p-normal. (b) The proof is similar to that of (a). Recall that a space X is pre-T2 [15] if every pair of distinct points in X are separated by disjoint preopen sets. An weakly Hausdorff space [6] is a topological space in which each point of X is an intersection of regular closed sets of X . A subset A of a space X is called S-closed relative to X [24] if every cover of A by semiopen sets of X has a finite subfamily whose closures cover A. Theorem 5.12. If f : X ! Y is a pre-gp-continuous injection and Y is weakly Hausdorff p-normal, then X is pre-T2 . Proof. Let x1 and x2 be distinct points in X . Then singletons ff ðxi Þg are obviously S-closed relative to Y for i ¼ 1; 2. By Lemma 2.2 in [13], ff ðxi Þg are disjoint closed sets of Y . Since Y is p-normal, there exist disjoint preopen sets V1 and V2 such that ff ðxi Þg Vi for i ¼ 1; 2. Since f is pre-pg-continuous, f 1 ðV1 Þ and f 1 ðV2 Þ are disjoint gp-open sets containing x1 and x2 , respectively. Hence X is pre-T2 . Theorem 5.13. If f : X ! Y is a pre-gp-closed surjection such that f 1 ðyÞ is S-closed relative to X for each y 2 Y and X is an weakly Hausdorff p-normal, then for each distinct points y1 and y2 in Y there exist gp-open sets V1 and V2 of Y such that yi 2 Vi for i ¼ 1; 2.
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Proof. Let y1 and y2 be any two distinct points in Y . Then by Lemma 2.2 in [13], f 1 ðy1 Þ and f 1 ðy2 Þ are disjoint closed subsets of X . Since X is p-normal, there exist disjoint preopen sets U1 and U2 such that f 1 ðyi Þ Ui for i ¼ 1; 2. Since f is pre-gp-closed, there exist gp-open sets V1 and V2 containing y1 and y2 , respectively, such that f 1 ðVi Þ Ui for i ¼ 1; 2. Since U1 and U2 are disjoint, V1 \ V2 ¼ /.
References [1] Arokiarani I, Balachandran K, Dontchev J. Some characterizations of gp-irresolute and gp-continuous maps between topological spaces. Mem Fac Sci Kochi Univ Ser A (Math) 1999;20:93–104. [2] Arya SP, Gupta R. On strongly continuous functions. Kyungpook Math J 1974;14:131–41. [3] Biswas N. On some mappings in topological spaces. Bull Calcutta Math Soc 1969;61:127–33. [4] Bourbaki N. General Topology Part I. Mass. Reading: Addison-Wesley; 1966. [5] Carnahan D. Some properties related to compactness in topological spaces. PhD thesis, University of Arkansas, 1973. [6] Dontchev J, Ganster M. On d-generalized closed sets and T3=4 -spaces. Mem Fac Sci Kochi Univ Ser A (Math) 1996;17:15–31. [7] Dontchev J, Ganster M, Noiri T. On p-closed spaces. Int J Math Math Sci 2000;24:203–12. [8] Dontchev J, Maki H. On behavior of gp-closed sets and their generalizations. Mem Fac Sci Kochi Univ Ser A (Math) 1998;19:57– 72. [9] Gnanambal Y. On generalized preregular closed sets in topological spaces. Indian J Pure Appl Math 1997;28:351–60. [10] Levine N. Generalized closed sets in topology. Rend Circ Mat Palermo 1970;19(2):89–96. [11] Maki H, Umehara J, Noiri T. Every topological space is pre-T1=2 . Mem Fac Sci Kochi Univ Ser A (Math) 1996;17:33–42. [12] Malghan SR, Navalagi GB. Almost-p-regular, p-completely regular and almost-p-completely regular spaces. Bull Math Soc Sci Math R S Roumanie 1990;34:317–26. [13] Mashhour AS, Abd El-Monsef ME, El-Deeb SN. On precontinuous and weak pre- continuous functions. Proc Math Phys Soc Egypt 1982;53:47–53. [14] Mashhour AS, Abd El-Monsef ME, Hasanein IA, Noiri T. Strongly compact spaces. Delta J Sci 1984;8:30–46. [15] Navalagi GB. Pre-neighbourhoods. Math Ed (Siwan) 1998;32:201–6. [16] Navalagi GB. P -normal, almost p-normal and mildly p-normal spaces. Topology Atlas Preprint #427. URL: http://at.yorku.ca/i/d/ e/b/71.htm. [17] Noiri T. Mildly normal spaces and some functions. Kyungpook Math J 1996;36:183–90. [18] Noiri T. Almost p-regular spaces and some functions. Acta Math Hungar 1998;79:207–16. [19] Noiri T, Maki H, Umehara J. Generalized preclosed functions. Mem Fac Sci Kochi Univ Ser A (Math) 1998;19:13–20. [20] Nour TM. Contributions to the theory of bitopological spaces. PhD thesis, Delhi University, India, 1989. [21] Palaniappan N, Rao KC. Regular generalized closed sets. Kyungpook Math J 1993;33:211–9. [22] Park JH, Park YB, Lee BY. On gp-closed sets and pre gp-continuous functions. Indian J Pure Appl Math 2002;33:3–12. [23] Popa V. Properties of H -almost continuous functions. Bull Math Soc Sci Math R S Roumanie 1987;31:163–8. [24] Reilly IL, Vamanamurthy MK. On a-continuity in topological spaces. Acta Math Hungar 1985;45:27–32. [25] Rose D. Weak openness and almost openness. Int J Math Math Sci 1984;7:35–40. [26] Singal MK, Singal AR. Almost-continuous mappings. Yokohama Math J 1968;16:63–73. [27] Singal MK, Singal AR. Mildly normal spaces. Kyungpook Math J 1973;13:27–31. [28] Velicko NY. H -closed topological spaces. Am Math Soc Transl 1968;78:103–18.
Almost p-normal, mildly p-normal spaces and some functions Jin Han Park Division of Mathematical Sciences, Pukyong National University, 599-1 Daeyeon 3-Dong Nam-Gu, Pusan 608-737, South Korea Accepted 14 January 2003
Abstract The aim of this paper is to obtain some characterizations of almost p-normal spaces and mildly p-normal spaces and to improve the preservation theorems of p-normal spaces and mildly p-normal spaces established by Navalagi [P normal, almost p-normal and mildly p-normal spaces. Topology Atlas Preprint #427. URL: http://at.yorku.ca/i/d/e/b/ 71.htm.]. Ó 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction and preliminaries In 1973, Singal and Singal [27] introduced a weak form of normal spaces called mildly normal spaces. In 1989, Nour [20] used preopen sets to define p-normal spaces. Recently, Navalagi [16] have continued the study of further properties of p-normal spaces and also defined and investigated mildly p-normal (resp. almost p-normal) spaces which are generalizations of both mildly normal (resp. almost normal) spaces and p-normal spaces. On the other hand, Levine [10] initiated the investigation of so-called g-closed sets in topological spaces, since then many modifications of g-closed sets were defined and investigated by a large number of topologists (see, References of this paper). In [11,19], Maki and coworkers defined and investigated the concept of gp-closed sets and used this notion to obtain a characterization of pnormal spaces. This notion is generalization of preclosed sets which were further studied by Dontchev and Maki [8], Arokiarani et al. [1], Noiri et al. [19] and Park et al. [22]. Recently, Gnanambal [9] have defined and studied the notion of gpr-closed sets which are implied by both that of rg-closed sets and gp-closed sets. Noiri [18] called gpr-closed set rgpclosed and used the sets to obtain further characterizations of almost p-regular spaces due to Malghan and Navalagi [12]. In this paper, we further study properties of gpr-closed sets and gpr-open sets and obtain new characterizations of almost p-normal spaces and mildly p-normal spaces by using gpr-open sets. We use pre-rgp-closed functions [18] to obtain certain preservation theorems of p-normal spaces and mildly p-normal spaces. The main result of this paper is that p-normality (resp. mild p-normality) is preserved under pre-gp-closed continuous (resp. pre-rgp-closed R-map) surjection. Throughout this paper, spaces ðX ; sÞ and ðY ; rÞ (or simply X and Y ) always mean topological spaces on which no separation axioms are assumed unless explicitly stated. Let A be a subset of a space X . The closure of A and the interior of A are denoted by clðAÞ and intðAÞ, respectively. A subset A is said to be regular open (resp. regular closed) if A ¼ intðclðAÞÞ (resp. A ¼ clðintðAÞÞ). A subset A is said to be preopen [13] if A intðclðAÞÞ. The complement of a preopen set is called preclosed. The intersection of all preclosed sets containing A is called the preclosure [13] of A and is denoted by pclðAÞ. Dually, the preinterior of A, denoted by pintðAÞ is defined to be the union of all preopen sets contained in A. A subset A is said to be a preneighborhood [15] of x if there exists a preopen set U such that x 2 U A.
E-mail address: [email protected] (J.H. Park). 0960-0779/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0960-0779(02)00650-1
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We recall the following definition used in sequel. Definition 1.1. A subset A of a space X is called: (a) generalized preclosed [11] (briefly, gp-closed) if pclðAÞ U whenever A U and U is open in X , (b) regular generalized closed [21] (briefly rg-closed) if clðAÞ U whenever A U and U is regular open in X , (c) generalized preregular closed [9] (briefly, gpr-closed), or regular generalized preclosed [18] if pclðAÞ U whenever A U and U is regular open in X , (e) gp-open (resp. rg-open, gpr-open) if the complement of A is gp-closed (resp. rg-closed, gpr-closed). 2. Generalized preregular closed sets We first recall following lemmas to obtain further results for gpr-closed sets. Lemma 2.1 [12]. If Y is an open subspace of a space X and A is a subset of Y , then pclY ðAÞ ¼ pclðAÞ \ ðY Þ. Lemma 2.2 [9]. If A is a regular open and gpr-closed subset of a space X , then A is preclosed in X . Theorem 2.3. Let Y be an open subspace of a space X and A Y . If A is gpr-closed in X , then A is gpr-closed in Y . Proof. Let U be a regular open set of Y such that A U . Then U ¼ V \ Y for some regular open set V of X . Since A is gpr-closed in X , we have pclðAÞ V and by Lemma 2.1, pclY ðAÞ ¼ pclðAÞ \ Y V \ Y ¼ U . Hence A is gpr-closed in Y. Theorem 2.4. Let Y be a gpr-closed and regular open subspace of a space X . If A is gpr-closed in Y , then A is gpr-closed in X . Proof. Let U be any regular open subset of X such that A U . Since U \ Y is regular open in Y and A is gpr-closed in Y , pclY ðAÞ U \ Y . By Lemmas 2.1 and 2.2, we have pclðAÞ ¼ pclðAÞ \ Y ¼ pclY ðAÞ U \ Y U. Hence A is gpr-closed in X . Corollary 2.5. If A is a gpr-closed regular open set and B is a preclosed set of a space X , then A \ B is gpr-closed. Theorem 2.6. Let A be a gpr-closed set. Then A ¼ pclðpintðAÞÞ if and only if pclðpintðAÞÞ n A is regular closed. Proof. If A ¼ pclðpintðAÞÞ, then pclðpintðAÞÞ n A ¼ / and hence pclðpintðAÞÞ n A is regular closed. Conversely, let pclðpintðAÞÞ n A be regular closed. Since pclðAÞ n A contains the regular closed set pclðpintðAÞÞ n A. By Theorem 3.15 in [9], pclðpintðAÞÞ n A ¼ / and hence pclðpintðAÞÞ ¼ A. A space X is said to be almost p-regular [12] if for each regular closed set A of X and each point x 2 X n A, there exist disjoint preopen sets U and V such that x 2 U and A V . A subset A of a space X is said to be p-closed relative to X [7] if for every cover fVa : a 2 Kg of A by preopen subsets of X , there exists a finite subset K0 of K such that A [fpclðVa Þ : a 2 K0 g. Theorem 2.7. If a space X is almost p-regular and a subset A of X is p-closed relative to X , then A is gpr-closed. Proof. Let U be any regular open set of X containing A. For each x 2 A, there exists a preopen set V ðxÞ such that x 2 V ðxÞ pclðV ðxÞÞ U . Since fV ðxÞ : x 2 Ag is a preopen cover of A, there exists a finite subset A0 of A such that A [fpclðV ðxÞÞ : x 2 A0 g. Hence we obtain A pclðAÞ [fpclðV ðxÞÞ : x 2 A0 g U . This shows that A is gpr-closed. Recall that a space X is called regular T1=2 [21] if every rg-closed set of X is regular closed. A set A of a space X is called d-closed [28] if A ¼ cld ðAÞ, where cld ðAÞ ¼ fx 2 X : intðclðU ÞÞ \ A 6¼ /; U is open set containing xg. Theorem 2.8. The following are equivalent for a space X in which every gpr-closed set is semiopen: (a) X is regular T1=2 . (b) Every singleton is regular closed or regular open in X .
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(c) Every singleton is regular closed or d-open in X . (d) Every gpr-closed subset of X is regular closed. Proof. (a) ) (b): Let x 2 X . If fxg is not regular closed, then A ¼ X n fxg is not regular open and hence A is trivially rgclosed. By the regular T1=2 property of X , A is regular closed and hence fxg is regular open. (b) () (c) follows from Lemma 4.2 in [6]. (b) ) (d): Let A X be gpr-closed. Let x 2 clðintððAÞÞ. We consider the following two cases: Case I. Let fxg be regular open. Since x 2 clðintððAÞÞ, then fxg \ intðAÞ 6¼ /. Thus x 2 A. Case II. Let fxg be regular closed. If we assume that x 62 A, then we have x 2 clðintðAÞÞ n A ¼ pclðAÞ n A (because pclðAÞ ¼ A [ clðintðAÞÞ), which is impossible according to Theorem 3.15 in [9]. Hence x 2 A. i.e., A is preclosed. Since A is gpr-closed, by hypothesis, A is semiopen and thus regular closed. (d) ) (a): Since every rg-closed is gpr-closed, the proof is clear. Now, we start with following lemma to get some results for gpr-open sets. Lemma 2.9 [9,16]. A subset A of a space X is gp-open (resp. gpr-open) if and only if F pintðAÞ whenever F is closed (resp. regular closed) and F A. Recall that two nonempty sets A and B of X are said to be preseparated [23] if pclðAÞ \ B ¼ / ¼ A \ pclðBÞ. Theorem 2.10. If A and B are gpr-open preseparated sets, then A [ B is gpr-open. Proof. Let F be a regular closed subset of A [ B. Then F \ pclðAÞ A and hence by Lemma 2.9, F \ pclðAÞ pintðAÞ. Similarly, F \ pclðBÞ pintðBÞ. Then we have F pintðA [ BÞ and hence A [ B is gpr-open. Theorem 2.11. If A is a gpr-open set of a space X , then U ¼ X whenever U is regular open and pintðAÞ [ ðX n AÞ U . Proof. Let U be a regular open set and pintðAÞ [ ðX n AÞ U. Then X n U ðX n pintðAÞ \ AÞ, i.e., ðX n U Þ pclðX n AÞ \ ðX n AÞ. Since X n A is gpr-closed, by Theorem 3.15 in [9], X n U ¼ /. Hence U ¼ X . The converse in the theorem above is not always true as shown by the following example. Example 2.12. Let X ¼ fa; b; c; dg and s ¼ fX ; /; fag; fbg; fa; bg; fa; b; cg; fa; b; dgg. Let A ¼ fb; c; dg. Then X is the only regular open set containing pintðAÞ [ ðX n AÞ but A is not gpr-closed in X . Theorem 2.13. Let Y be a regular open and gpr-closed subspace of a space X . If A is gpr-open in Y , then A is gpr-open in X . Proof. Let F be any regular closed set and F A. Since F is regular closed in Y and A is gpr-open in Y , F pintY ðAÞ and then F pintðAÞ \ Y . Hence F pintðAÞ and so A is gpr-open in X . 3. Characterizations of almost p-normal and mildly p-normal spaces Definition 3.1. A space X is said to be (a) p-normal [16], or prenormal [20] (resp. mildly p-normal [16]) if for every pair of disjoint closed (resp. regular closed) sets A and B of X , there exist disjoint preopen sets U and V such that A U and B V . (b) almost p-normal [16] if for each closed set A and regular closed set B of X such that A \ B ¼ /, there exist disjoint preopen sets U and V such that A U and B V . Theorem 3.2. The following are equivalent for a space X : (a) X is almost p-normal. (b) For each closed set A and regular closed set B such that A \ B ¼ /, there exist disjoint gp-open sets U and V such that A U and B V . (c) For each closed set A and regular closed set B such that A \ B ¼ /, there exist disjoint gpr-open sets U and V such that A U and B V .
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(d) For each closed set A and each regular open set B containing A, there exists a gpr-open set V A V pclðV Þ B. (e) For each rg-closed set A and each regular open set B containing A, there exists a preopen set V clðAÞ V pclðV Þ B. (f) For each rg-closed set A and each regular open set B containing A, there exists a gp-open set V clðAÞ V pclðV Þ B. (g) For each rg-closed set A and each regular open set B containing A, there exists a gpr-open set V clðAÞ V pclðV Þ B.
of X such that of X such that of X such that of X such that
Proof. It is obvious that (a) ) (b) ) (c) and (e) ) (f) ) (g) ) (d). (c) ) (d): Let A be a closed set and B be a regular open subset of X containing A. There exist disjoint gpr-open sets V and W such that A V and X n B W . By Lemma 2.9, we have X n B pintðW Þ and V \ pintðW Þ ¼ /. Therefore, we obtain pclðV Þ \ pintðW Þ ¼ / and hence A V pclðV Þ X n pintðW Þ B. (d) ) (a): Let A be any closed set and B be any regular closed set such that A \ B ¼ /. Then X n B is a regular open set containing A and there exists a gpr-open set G of X such that A G pclðGÞ X n B. Put U ¼ pintðGÞ and V ¼ X n pclðGÞ. Then U and V are disjoint preopen sets of X such that A U and B V . Hence X is almost p-normal. (a) ) (e): Let A be a rg-closed set and B be a regular open subset of X containing A. Then clðAÞ B and by Theorem 4.3 in [16], there exists a preopen set V such that clðAÞ V pclðV Þ B. Remark 3.3. We can obtain characterizations of almost p-normal spaces by replacing ‘‘A is rg-closed’’ in (e)–(g) of Theorem 3.2 with ‘‘A is g-closed’’. The proof is similar to that of Theorem 3.2. Theorem 3.4. The following are equivalent for a space X : (a) (b) (c) (d)
X is mildly p-normal. For any disjoint regular closed sets A and B of X , there exist disjoint gp-open sets U and V such that A U and B V . For any disjoint regular closed sets A and B of X , there exist disjoint gpr-open sets U and V such that A U and B V . For each regular closed set A and each regular open set B containing A, there exists a gp-open set V of X such that A V pclðV Þ B. (e) For each regular closed set A and each regular open set B containing A, there exists a gpr-open set V of X such that A V pclðV Þ B.
Proof. The proof is similar to that of Theorem 3.2.
4. Pre-gpr-continuous and gpr-irresolute functions Definition 4.1. A function f : X ! Y is called: (a) precontinuous [13] (resp. preirresolute [24]) if f 1 ðF Þ is preclosed in X for every closed (resp. preclosed) set F of Y ; (b) gp-continuous [1] (resp. pre-gp-continuous [22], gp-irresolute [1]) if f 1 ðF Þ is gp-closed in X for every closed (resp. preclosed, gp-closed) set F of Y ; (c) gpr-continuous [9] (resp. pre-gpr-continuous, gpr-irresolute [9]) if f 1 ðF Þ is gpr-closed) in X for every closed (resp. preclosed, gpr-closed) set F of Y . From the definition stated above, we obtain the following diagram of implications: gp-irresoluteness gpr-irresoluteness + + preirresoluteness ) pre-gp-continuity ) pre-gpr-continuity + + + precontinuity ) gp-continuity ) gpr-continuity Remark 4.2 (a) None of the implications in above diagram are reversible; (b) Preirresoluteness, gp-irresoluteness and gpr-irresoluteness are mutually independent; (c) Pre-gpr-continuity and gp-continuity are independent of each other.
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Example 4.3 (a) Let X ¼ fa; b; c; dg, s ¼ fX ; /; fag; fbg; fa; bgg and r ¼ fX ; /; fc; dgg. Let f : ðX ; sÞ ! ðX ; rÞ be an identity. Then f is gpr-continuous but not pre-gpr-continuous since fag is preclosed in ðX ; rÞ and f 1 ðfagÞ is not gpr-closed in ðX ; sÞ. (b) Let ðX ; sÞ and ðX ; rÞ be spaces given in (a). Let f : ðX ; sÞ ! ðX ; rÞ be a function defined by f ðaÞ ¼ f ðbÞ ¼ a, f ðcÞ ¼ b and f ðdÞ ¼ d. Then f is gpr-irresolute but not pre-gp-continuous since fag is preclosed in ðX ; rÞ and f 1 ðfagÞ is not gp-closed in ðX ; sÞ. (c) Let X ¼ fa; b; c; dg, s ¼ fX ; /; fag; fbg; fa; bgg and r ¼ fX ; /; fag; fa; bgg. Let f : ðX ; sÞ ! ðX ; rÞ be a function defined by f ðaÞ ¼ a, f ðbÞ ¼ f ðcÞ ¼ c and f ðdÞ ¼ d. Then f is gp-irresolute (and hence pre-gpr-continuous) but not gpr-irresolute since fag is gpr-closed in ðX ; rÞ and f 1 ðfagÞ is not gpr-closed in ðX ; sÞ. (d) Let X ¼ fa; b; c; dg, s ¼ fX ; /; fag; fa; bg; fc; dg; fa; c; dgg and r ¼ fX ; /; fbg; fa; cg; fa; b; cgg. Let f : ðX ; sÞ ! ðX ; rÞ be an identity. Then f is pre-gpr-continuous but not gp-continuous since fa; c; dg is closed in ðX ; rÞ and f 1 ðfa; c; dgÞ is not gp-closed in ðX ; sÞ. (e) Let ðX ; sÞ and ðX ; rÞ be spaces given in (d). Let f : ðX ; rÞ ! ðX ; sÞ be a function defined by f ðaÞ ¼ a, f ðbÞ ¼ c, f ðcÞ ¼ b and f ðdÞ ¼ d. Then f is gp-continuous but not pre-gpr-continuous since fcg is preclosed in ðX ; sÞ and f 1 ðfcgÞ is not gpr-closed in ðX ; rÞ. (f) Let X ¼ fa; b; c; dg, s ¼ fX ; /; fag; fb; cg; fa; b; cgg and r ¼ fX ; /; fag; fbg; fa; bg; fa; b; cg; fa; b; dgg. Let f : ðX ; sÞ ! ðX ; rÞ be an identity. Then f is preirresolute but not gpr-irresolute since fb; cg is gpr-closed in ðX ; rÞ and f 1 ðfb; cgÞ is not gpr-closed in ðX ; sÞ. Recall that a space X is called preregular T1=2 [9] if every gpr-closed set is preclosed. Theorem 4.4. If a function f : X ! Y is gpr-continuous (resp. pre-gpr-continuous) and X is preregular T1=2 , then f is precontinuous (resp. preirresolute). Proof. Let F be any closed (resp. preclosed) set of Y . Since f is gpr-continuous (resp. pre-gpr-continuous), f 1 ðF Þ is gprclosed in X and then f 1 ðF Þ is preclosed in X . Hence f is precontinuous (resp. preirresolute). A space X is called submaximal [4] if every dense subset of X is open. Note that the submaximal spaces are exactly the space where every preopen set is open. Theorem 4.5. If a function f : X ! Y is pre-gpr-continuous (resp. gpr-continuous) and Y is preregular T1=2 (resp. submaximal), then f is gpr-irresolute (resp. pre-gpr-continuous). Proof. Let F be any gpr-closed (resp. preclosed) subset of Y . Since Y is preregular T1=2 (resp. submaximal), then F is preclosed (resp. closed) in Y . Hence f 1 ðF Þ is rgp-closed in X . This show that f is gpr-irresolute (resp. pre-gpr-continuous). Theorem 4.6. Let f : X ! Y and g : Y ! Z be functions. (a) If f is gpr-irresolute and g is gpr-continuous (resp. pre-gpr-continuous), then the composition g f : X ! Z is gpr-continuous (resp. pre-gpr-continuous). (b) If f and g are pre-gpr-continuous and Y is preregular T1=2 , then the composition g f : X ! Z is pre-gpr-continuous. (c) If f and g are gpr-irresolute, then the composition g f : X ! Z is gpr-irresolute. The composition of two gpr-continuous functions need not be pre-gpr-continuous as seen by the following example. Example 4.7. Let X ¼ fa; b; c; dg, s ¼ fX ; /; fag; fb; cg; fa; b; cgg, r ¼ fX ; /; fag; fbg; fa; bgg and q ¼ fX ; /; fcdgg. Let f : ðX ; sÞ ! ðX ; rÞ be a function defined by f ðaÞ ¼ f ðdÞ ¼ d, f ðbÞ ¼ b and f ðcÞ ¼ a and g : ðX ; rÞ ! ðX ; qÞ be a function defined by gðaÞ ¼ gðbÞ ¼ a, gðdÞ ¼ c and gðdÞ ¼ d. Then f and g are pre-gpr-continuous but the composition g f is not pre-gpr-continuous since fag is preclosed in ðX ; qÞ and ðg f Þ1 ðfagÞ is not gpr-closed in ðX ; sÞ. 5. Preservation theorems Definition 5.1. A function f : X ! Y is called (a) almost continuous [26] (resp. R-map [5], completely continuous [2]) if f 1 ðV Þ is open (resp. regular open, regular open) in X for every regular open (resp. regular open, open) set V of Y ,
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(b) almost preirresolute [16] if for each x 2 X and preneighborhood V of f ðxÞ, pclðf 1 ðV ÞÞ is a preneighborhood of x, (c) pre-gp-closed [19] (resp. pre-rgp-closed [18], M-preclosed [14]) if f ðF Þ is gp-closed (resp. gpr-closed, preclosed) in Y for every preclosed set F of X , (d) rc-preserving [17] (resp. almost closed [26]) if f ðF Þ is regular closed (resp. closed) in Y for every regular closed set F of X , (e) preopen [13], or almost open [25] (resp. M-preopen [14], semiopen [3]) if f ðU Þ is preopen (resp. preopen, semiopen) in Y for every open (preopen, open) set U of X , (f) weakly open [25] if f ðU Þ intðf ðclðU ÞÞÞ for every open set U of X . Note that preopeness and weak openness are independent of each other [25]. Lemma 5.2 [19]. A surjection f : X ! Y is pre-gp-closed (resp. pre-rgp-closed) if and only if for each subset B of Y and each preopen set U of X containing f 1 ðBÞ there exists a gp-open (resp. gpr-open) set of V of Y such that B V and f 1 ðV Þ U . Lemma 5.3 [19]. If a function f : X ! Y is continuous preopen, then f is M-preopen. Lemma 5.4. If a function f : X ! Y is weakly open continuous, then f is M-preopen and R-map. Proof. Since every weakly open continuous function is R-map [18], we show that f is preopen. Let U be any open set of X . Since f is weakly open and continuous, f ðU Þ f ðIntðClðU ÞÞÞ Intðf ðClðIntðClðU ÞÞÞÞÞ Intðf ðClðU ÞÞÞ IntðClðf ðU ÞÞÞ: Then f ðU Þ is preopen in Y and thus f is preopen. Hence, by Lemma 5.3, f is M-preopen.
Lemma 5.5. If a function f : X ! Y is semiopen precontinuous, then f is preirresolute. Proof. Let V be a preopen set of Y . Then sclðV Þ ¼ intðclðV ÞÞ. Since f is semiopen, f 1 ðintðclðV ÞÞÞ ¼ f 1 ðsclðV ÞÞ clðf 1 ÞÞ. Since f is precontinuous, we have f 1 ðV Þ f 1 ðintðclðV ÞÞÞ intðclðf 1 ðintðclðV ÞÞÞÞÞ intðclðf 1 ðV ÞÞÞ: Thus, f 1 ðV Þ is preopen and hence f is preirresolute.
Theorem 5.6 (a) If f : X ! Y is a preopen continuous almost preirresolute surjection and X is p-normal, then Y is p-normal. (b) If f : X ! Y is an R-map preopen continuous almost preirresolute surjection and X is almost p-normal, then Y is almost p-normal. Proof (a) Let A be a closed subset of Y and B be an open set containing A. Then f 1 ðAÞ is closed and f 1 ðBÞ is an open set of X such that f 1 ðAÞ f 1 ðBÞ. By the p-normality of X , there exists a preopen set U of X such that f 1 ðAÞ U pclðU Þ f 1 ðBÞ and then f ðf 1 ðAÞÞ f ðU Þ f ðpclðU ÞÞ f ðf 1 ðBÞÞ. Since f is preopen continuous almost preirresolute surjection, by Lemma 5.3 we have f ðU Þ is preopen set of Y such that A f ðU Þ pclðf ðU ÞÞ B. Hence Y is p-normal. (b) The proof is entirely analogous to (a). Corollary 5.7 (a) [16, Theorem 3.9] If f : X ! Y is an M-preopen continuous almost preirresolute function from a p-normal space X onto a space Y , then Y is p-normal. (b) [16, Theorem 4.6] If f : X ! Y is continuous M-preopen R-map and almost preirresolute surjection from an almost pnormal space X onto a space Y , then Y is almost p-normal. Theorem 5.8 (a) If f : X ! Y is an weakly open continuous almost preirresolute surjection and X is p-normal, then Y is p-normal. (b) If f : X ! Y is a preopen semiopen continuous surjection and X is a p-normal space, then Y is p-normal.
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Proof (a) Let A be a closed subset of Y and B be an open set containing A. Then f 1 ðAÞ is closed and f 1 ðBÞ is an open set of X such that f 1 ðAÞ f 1 ðBÞ. By the p-normality of X , there exists a preopen set U of X such that f 1 ðAÞ U pclðU Þ f 1 ðBÞ and then f ðf 1 ðAÞÞ f ðU Þ f ðpclðU ÞÞ f ðf 1 ðBÞÞ. Since f is weakly open continuous almost preirresolute surjection, by Lemma 5.4 we have f ðU Þ is preopen set of Y such that A f ðU Þ pclðf ðUÞÞ B. Hence Y is p-normal. (b) The proof follows from Theorem 5.6(a) using Lemmas 5.3 and 5.5. Theorem 5.9 (a) If f : X ! Y (b) If f : X ! Y (c) If f : X ! Y (d) If f : X ! Y (e) If f : X ! Y mal.
is a pre-gp-closed continuous surjection and X is p-normal, then Y is p-normal. is an R-map pre-rgp-closed surjection and X is mildly p-normal, then Y is mildly p-normal. is a completely continuous pre-rgp-closed surjection and X is mildly p-normal, then Y is p-normal. is an almost continuous pre-rgp-closed surjection and X is p-normal, then Y is mildly p-normal. is a continuous weakly open pre-rgp-closed surjection and X is mildly p-normal, then Y is almost p-nor-
Proof (a) Let A and B be any disjoint closed sets of Y . Then f 1 ðAÞ and f 1 ðBÞ are disjoint closed sets of X . Since X is p-normal, there exist disjoint preopen sets U and V such that f 1 ðAÞ U and f 1 ðBÞ V . By Lemma 5.2, there exist gp-open sets G and H of Y such that A G, B H, f 1 ðGÞ U and f 1 ðH Þ V . Since U and V are disjoint, G and H are disjoint. By Lemma 2.9, we have A pintðGÞ, B pintðH Þ and pintðGÞ \ pintðHÞ ¼ /. This shows that Y is p-normal. The proofs of (b)–(e) are entirely analogous to (a). Corollary 5.10 (a) [16, Theorem 3.11] If f : X ! Y is an M-preclosed continuous function from a p-normal space onto a space Y , then Y is p-normal. (b) [16, Theorem 5.9] If f : X ! Y is an R-map M-preclosed function from a mildly p-normal space X onto a space Y , then Y is mildly p-normal. Theorem 5.11 (a) If f : X ! Y is a pre-gpr-continuous rc-preserving injection and Y is mildly p-normal, then X is mildly p-normal. (b) If f : X ! Y is a pre-gpr-continuous almost closed injection and Y is p-normal, then X is mildly p-normal. Proof (a) Let A and B be any disjoint regular closed sets of X . Since f is re-preserving injection, f ðAÞ and f ðBÞ are disjoint regular closed sets of Y . By mild p-normality of Y , there exist disjoint preopen sets U and V of Y such that f ðAÞ U and f ðBÞ V . Since f pre-gpr-continuous, f 1 ðU Þ and f 1 ðV Þ are disjoint gpr-open sets containing A and B, respectively. Hence by Theorem 3.4, X is mildly p-normal. (b) The proof is similar to that of (a). Recall that a space X is pre-T2 [15] if every pair of distinct points in X are separated by disjoint preopen sets. An weakly Hausdorff space [6] is a topological space in which each point of X is an intersection of regular closed sets of X . A subset A of a space X is called S-closed relative to X [24] if every cover of A by semiopen sets of X has a finite subfamily whose closures cover A. Theorem 5.12. If f : X ! Y is a pre-gp-continuous injection and Y is weakly Hausdorff p-normal, then X is pre-T2 . Proof. Let x1 and x2 be distinct points in X . Then singletons ff ðxi Þg are obviously S-closed relative to Y for i ¼ 1; 2. By Lemma 2.2 in [13], ff ðxi Þg are disjoint closed sets of Y . Since Y is p-normal, there exist disjoint preopen sets V1 and V2 such that ff ðxi Þg Vi for i ¼ 1; 2. Since f is pre-pg-continuous, f 1 ðV1 Þ and f 1 ðV2 Þ are disjoint gp-open sets containing x1 and x2 , respectively. Hence X is pre-T2 . Theorem 5.13. If f : X ! Y is a pre-gp-closed surjection such that f 1 ðyÞ is S-closed relative to X for each y 2 Y and X is an weakly Hausdorff p-normal, then for each distinct points y1 and y2 in Y there exist gp-open sets V1 and V2 of Y such that yi 2 Vi for i ¼ 1; 2.
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Proof. Let y1 and y2 be any two distinct points in Y . Then by Lemma 2.2 in [13], f 1 ðy1 Þ and f 1 ðy2 Þ are disjoint closed subsets of X . Since X is p-normal, there exist disjoint preopen sets U1 and U2 such that f 1 ðyi Þ Ui for i ¼ 1; 2. Since f is pre-gp-closed, there exist gp-open sets V1 and V2 containing y1 and y2 , respectively, such that f 1 ðVi Þ Ui for i ¼ 1; 2. Since U1 and U2 are disjoint, V1 \ V2 ¼ /.
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