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Almost strongly θ-precontinuous functions
Chaos, Solitons and Fractals 28 (2006) 32–41 www.elsevier.com/locate/chaos
Almost strongly h-precontinuous functions
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Jin Han Park *, Sang Wook Bae, Yong Beom Park Division of Mathematical Sciences, Pukyong National University, 599-1 Daeyeon 3-Dong, Nam-Gu, Pusan 608-737, South Korea Accepted 6 May 2005
Abstract In this paper, we introduce a new class of functions called almost strongly h-precontinuous function which is a generalization of almost strongly h-continuous functions due to Noiri and Kang [Noiri T, Kang SM. On almost strongly hcontinuous functions. Indian J Pure Appl Math 1984;15(1):1–8] and strongly h-precontinuous functions due to Noiri [Noiri T. Strongly h-precontinuous functions. Acta Math Hung 2001;90(4):307–16]. Some characterizations and several properties concerning almost strongly h-precontinuous functions are obtained. The relationships between almost strongly h-precontinuity and other types of continuity are also given. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Relation of topology and physics have been appeared in [7–9,35], i.e. topology plays a significant role in quantum physics, high energy physics and superstring theory. Continuity on topological spaces, as important and basic subject in study of topology, have been researched by many mathematicians. This concept has been generalized by weaker forms of open sets such as a-open sets [21], semi-open sets [14], pre-open sets [17], b-open sets [3] and b-open sets [1]. In 1984, Mashhour et al. [17] introduced and investigated the notion of precontinuous functions. Precontinuity was called near continuity by Pta´k [29] and also almost continuity by Frolı´k [11] and Husain [12]. Jankovic´ [13] introduced almost weak continuity as a weak form of precontinuity. Popa and Noiri [27] introduced weak precontinuity and showed that almost weak continuity is equivalent to weak precontinuity. Recently, Noiri [23] introduced and investigated the notion of strongly h-precontinuous functions which is implied by that of strongly h-continuous functions [15] and implies that of precontinuous functions. In this paper, we introduce a new class of functions called almost strongly h-precontinuous functions which is contained in the class of weakly precontinuous functions and contains both the class of almost strongly h-continuous functions [24] and the class of strongly h-precontinuous functions. We investigate almost strongly h-precontinuous functions and obtain several improvements of results established by Noiri [23]. It is also shown that every almost strongly h-precontinuous surjective image of p-closed (resp. countably p-closed) space is nearly compact (resp. nearly countably compact).
q *
This work was supported by grant no. R05-2002-000-01045-0 from the Korea Science & Engineering Foundation. Corresponding author. E-mail address: [email protected] (J.H. Park).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.05.058
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2. Preliminaries Throughout this paper, spaces (X, s) and (Y, r) (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. For a subset A of (X, s), cl(A) and int(A) represent the closure of A and the interior of A with respect to s, respectively. A subset A is said to be regular open (resp. regular closed) if A = int(cl(A)) (resp. A = cl(int(A))). A point x of X is called a h-cluster [36] (resp. d-cluster [36]) point of A if cl(U) \ A 5 ; (resp. int(cl(U)) \ A 5 ;) for every open set U of X containing x. The set of all h-cluster (resp. d-cluster) points of A is called the h-closure [36] (resp. d-closure [36]) of A and is denoted by h-cl(A) (resp. dcl(A)). A subset A is said to be h-closed [36] (resp. d-closed [36]) if h-cl(A) = A (resp. d-cl(A) = A). The complement of a h-closed (resp. d-closed) set is said to be h-open (resp. d-open). A subset A is said to be pre-open [17] (resp. a-open [21], semi-open [14], b-open [1]) if A int(cl(A)) (resp. A int(cl(int(A))), A cl(int(A)), A cl(int(cl(A)))). The complement of a pre-open (resp. a-open, semi-open, b-open) set is said to be preclosed (resp. a-closed, semi-closed, b-closed). The family of all pre-open sets of X is denoted by PO(X) and the family {U 2 PO(X) : x 2 U} is denoted by PO(X, x), where x is a point of X. The intersection of all preclosed sets of X containing A is called the preclosure [6] of A and is denoted by pcl(A). The a-closure, semi-closure and b-closure are similarly defined and are denoted by a-cl(A), scl(A) and b-cl(A) [5,20]. The union of all pre-open sets of X contained in A is called preinterior and is denoted by pint(A). A point x of X is called a pre-h-cluster point of A if pcl(U) \ A 5 ; for every pre-open set U of X containing x. The set of all pre-h-cluster points of A is called the pre-hclosure of A and is denoted by h-pcl(A). A subset A is said to be pre-h-closed [26] if A = h-pcl(A). The complement of a pre-h-closed set is said to be pre-h-open. Definition 2.1. A function f : X ! Y is said to be (a) almost continuous [34] (briefly, a.c.S.) if for each x 2 X and each open set V of Y containing f(x), there exists an open set U containing x such that f(U) int(cl(V)); (b) d-continuous [22] if for each x 2 X and each open set V of Y containing f(x), there exists an open set U containing x such that f(int(cl(U))) int(cl(V)); (c) precontinuous [17] or almost continuous [12] if for each x 2 X and each open set V of Y containing f(x), there exists U 2 PO(X, x) such that f(U) V; (d) weakly precontinuous [27] or almost weakly continuous [13] if for each x 2 X and each open set V of Y containing f(x), there exists U 2 PO(X, x) such that f(U) cl(V); (e) strongly h-precontinuous [23] (briefly, st.h.p.c.) if for each x 2 X and each open set V of Y containing f(x), there exists U 2 PO(X, x) such that f(pcl(U)) V. Definition 2.2. A function f : X ! Y is said to be almost strongly h-precontinuous (briefly, a.st.h.p.c.) if for each x 2 X and each open set V of Y containing f(x), there exists U 2 PO(X, x) such that f(pcl(U)) int(cl(V)). Definition 2.3. A function f : X ! Y is said to be strongly h-continuous [22] (resp. almost strongly h-continuous [24] (briefly, a.st.h-c.)) if for each x 2 X and each open set V of Y containing f(x), there exists an open neighborhood U of x such that f(cl(U)) V (resp. f(cl(U)) int(cl(V))). Remark 2.4. Almost strongly h-precontinuity is implied by both almost strongly h-continuity and strongly h-precontinuity and implies weak precontinuity. None of these implications is reversible as the following examples show. Moreover, the following Examples 2.5 and 2.7 show that almost strongly h-precontinuity and continuity are independent of each other. Example 2.5. Let X = {a, b, c, d}, s = {X, ;, {c}, {a, b}, {a, b, c}} and r = {X, ;, {a}, {c}, {a, b}, {a, c}, {a, b, c}, {a, c, d}}. Define a function f : (X, s) ! (X, r) as follows: f(a) = f(b) = b and f(c) = f(d) = a. Then f is a.st.h.p.c. (even a.st.h-c.) but it is neither continuous nor st.h.p.c. Example 2.6. Let (X, s) and (X, r) be the spaces in Example 2.5. Define a function f : (X,s) ! (X,r) as follows: f(a) = b and f(b) = f(c) = f(d) = c. Then f is a.st.h.p.c. but it is not a.st.h-c. Example 2.7. Let (X, s) be the space in Example 2.5. Then the identity function f : (X, s) ! (X, s) is continuous (hence weakly precontinuous) but not a.st.h.p.c.
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3. Characterizations Theorem 3.1. For a function f : X ! Y, the following are equivalent: (a) (b) (c) (d) (e) (f) (g) (h)
f is a.st.h.p.c.; f1(V) is pre-h-open in X for each regular open set V of Y; f1(F) is pre-h-closed in X for each regular closed set F of Y; for each x 2 X and each regular open set V of Y containing f(x), there exists U 2 PO(X, x) such that f(pcl(U)) V; f1(V) is pre-h-open in X for each d-open set V of Y; f1(F) is pre-h-closed in X for each d-closed set F of Y; f(h-pcl(A)) d-cl(f(A)) for each subset A of X; h-pcl(f1(B)) f1(d-cl(B)) for each subset B of Y.
Proof (a) ) (b): Let V be any regular open set of Y and x 2 f1(V). There exists U 2 PO(X, x) such that f(pcl(U)) V. Therefore, we have x 2 U pcl(U) f1(V). This shows that f1(V) is pre-h-open in X. (b) ) (c): Let F be any regular closed set of Y. By (b), f1(F) = X f1(Y F) is pre-h-closed in X. (c) ) (d): Let x 2 X and V be any regular open set of Y containing f(x). By (c), f1(Y V) = X f1(V) is pre-hclosed in X. Since f1(V) is a pre-h-open set containing x, there exists U 2 PO(X, x) such that pcl(U) f1(V). Therefore, we have f(pcl(U)) V. (d) ) (e): Let V be any d-open set of Y and x 2 f1(V). There exists a regular open set G of Y such that f(x) 2 G V. By (d), there exists U 2 PO(X, x) such that f(pcl(U)) G. Therefore, we obtain x 2 U pcl(U) f1(V). This shows that f1(V) is pre-h-open in X. (e) ) (f): Let F be any d-closed set of Y. By (e), f1(F) = X f1(Y F) is pre-h-closed in X. (f) ) (g): Let A be any subset of Y. Since d-cl(f(A)) is d-closed in Y, f1(d-cl(f(A))) is pre-h-closed in X. Let x 62 f1(d-cl(f(A))). There exists U 2 PO(X, x) such that pcl(U) \ f1(d-cl(f(A))) = ; and thus pcl(U) \ A = ;. Hence x 62 h-pcl(A). Therefore, we have f(h-pcl(A)) d-cl(f(A)). (g) ) (h): Let B be any subset of Y. By (g), we have f(h-pcl(f1(B))) d-cl(B) and hence h-pcl(f1(B)) f1(d-cl(B)). (h) ) (a): Let x 2 X and V be any open set of Y containing f(x). Then G = Y int(cl(V)) is regular closed and hence d-closed in Y. By (h), h-pcl(f1(G)) f1(G) and hence f1(G) is pre-h-closed in X. Therefore, f1(int(cl(V))) is pre-hopen set containing x. There exists U 2 PO(X, x) such that pcl(U) f1(int(cl(V))). Therefore, we obtain f(pcl(U)) int(cl(V)). This shows that f is a.st.h.p.c. h It is known that the family of all d-open sets in a space (X,s) form a topology for X which is denoted by sd. However, sd is identical with the semiregularization ss of s and hence we use ss in the place of sd. For simplicity, we shall denote (X,ss) by Xs. Lemma 3.2 (Andrijevic´ [2]). scl(V) = int(cl(V)) for each pre-open set V of a space X. Theorem 3.3. For a function f : X ! Y the following are equivalent: (a) (b) (c) (d)
f is a.st.h.p.c.; for each x 2 X and each open set V of Y containing f(x), there exists U 2 PO(X, x) such that f(pcl(U)) scl(V); f1(V) h-pcl(f1(int(cl(V)))) for each open set V of Y; f : X ! Ys is st.h.p.c.
Proof (a) () (b): It follows from Lemma 3.2. (a) ) (c): Let V be any open set of Y and x 2 f1(V). By (a), there exists U 2 PO(X, x) such that f(pcl(U)) int(cl(V)). Therefore, we have x 2 U pcl(U) f1(int(cl(V))) and hence x 2 h-pcl(f1(int(cl(V)))). It follows that f1(V) h-pcl(f1(int(cl(V)))). (c) ) (d): Let x 2 X and V be any open set of Ys containing f(x). There exists a regular open set G of Y such that f(x) 2 G V. By (c), we have x 2 f1(G) h-pcl(f1(G)) and hence there exists U 2 PO(X) such that x 2 U pcl(U) f1(G). Therefore, we obtain f(pcl(U)) V. This shows that f : X ! Ys is st.h.p.c.
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(d) ) (a): Let V be regular open set of Y. For any x 2 f1(V), f(x) V and V is open in Ys. There exists U 2 PO(X, x) such that f(pcl(U)) V and hence pcl(U) f1(V). Therefore, we have f1(V) h-pcl(f1(V)) and f1(V) is pre-h-open in X. By Theorem 3.1, f is a.st.h.p.c. h
Lemma 3.4 [24]. For a subset V of a space X, then following hold: (a) a-cl(V) = cl(V) for each b-open set V of X. (b) pcl(V) = cl(V) for each semi-open set V of X.
Theorem 3.5. For a function f : X ! Y, the following are equivalent: (a) (b) (c) (d) (e)
f is a.st.h.p.c.; h-pcl(f1(V)) f1(cl(V)) for each b-open set V of Y; h-pcl(f1(V)) f1(cl(V)) for each semi-open set V of Y; h-pcl(f1(V)) f1(a-cl(V)) for each b-open set V of Y; h-pcl(f1(V)) f1(pcl(V)) for each semi-open set V of Y.
Proof (a) ) (b): Let V be any b-open set of Y. Then by Theorem 2.4 in [2], cl(V) is regular closed in Y. Since f is a.st.h.p.c., f1(cl(V)) is pre-h-closed in X and hence h-pcl(f1(V)) f1(cl(V)). (b) ) (c): This is obvious since every semi-open set is b-open. (c) ) (a): Let F be any regular closed set of Y. Then F is semi-open in Y and hence hpcl(f1(F)) f1(cl(F)) = f1(F). This shows that f1(F) is pre-h-closed in X. Therefore, f is a.st.h.p.c. (b) () (d): It follows from Lemma 3.4(a). (c) () (e): It follows from Lemma 3.4(b). h Recall that a space X is said to be almost regular [32] (resp. semi-regular) if for any regular open (resp. open) set U of X and each point x 2 U, there exits a regular open set V of X such that x 2 V cl(V) U (resp. x 2 V U). Theorem 3.6. Let f : X ! Y be a function. Then, the following properties hold: (a) If f is precontinuous and Y is almost regular, then f is a.st.h.p.c. (b) If f is a.st.h.p.c. and Y is semi-regular, then f is st.h.p.c.
Proof (a) Let x 2 X and V be any regular open set of Y containing f(x). Since Y is almost regular, there exists an open set W such that f(x) 2 W cl(W) V. Since f is precontinuous, there exists U 2 PO(X, x) such that f(U) W. We shall show that f(pcl(U)) cl(W). Suppose that y 62 cl(W). There exists an open neighborhood G of y such that G \ W = ;. Since f is precontinuous, f1(G) 2 PO(X) and f1(G) \ U = ; and hence f1(G) \ pcl(U) = ;. Therefore, we obtain G \ f(pcl(U)) = ; and y 62 f(pcl(U)). Consequently, we have f(pcl(U)) cl(W) V. (b) Let x 2 X and V be any open set of Y containing f(x). Since Y is semi-regular, there exists a regular open set W such that f(x) W V. Since f is a.st.h.p.c., there exists U 2 PO(X, x) such that f(pcl(U)) W. Therefore, we have f(pcl(U)) V. h Corollary 3.7. Let Y be a regular space. Then, the following properties are equivalent for a function f : X ! Y: (a) (b) (c) (d)
f f f f
is is is is
weakly precontinuous; precontinuous; a.st.h.p.c.; st.h.p.c.
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Proof. It follows from Theorem 3.2 of [23] and Theorem 3.6.
h
Definition 3.8. A space X is said to be p-regular [6] (resp. almost p-regular [16]) if for each closed (resp. regular closed) set F and each point x 2 X F, there exist disjoint pre-open sets U and V such that x 2 U and F V. Theorem 3.9 (a) If continuous function f : X ! Y is a.st.h.p.c., then X is almost p-regular. (b) If f : X ! Y is a.c.S. (resp. d-continuous) and X is p-regular (resp. almost p-regular), then f is a.st.h.p.c. Proof (a) Let f : X ! X be the identity. Then f is continuous and hence a.st.h.p.c. For any regular open set U of X and any point x of U, we have f(x) = x 2 U and there exists G 2 PO(X, x) such that f(pcl(G)) U. Therefore, we have x 2 G pcl(G) U and hence X is almost p-regular. (b) Suppose that f : X ! Y is a.c.S. (resp. d-continuous) and X is p-regular (resp. almost p-regular). For each x 2 X and any regular open set V containing f(x), f1(V) is an open (resp. regular open) set of X containing x. Since X is p-regular (resp. almost p-regular), there exists U 2 PO(X, x) such that x 2 U pcl(U) f1(V). Therefore, we have f(pcl(U)) V. This shows that f is a.st.h.p.c. h A space X is said to be submaximal [31] if each dense subset of X is open in X. It is shown in [31] that a space X is submaximal if and only if every pre-open set of X is open. Theorem 3.10. Let X be a submaximal space. Then f : X ! Y is a.st.h.p.c. if and only if f is a.st.h-c. Proof. Suppose that f is a.st.h.p.c. Let x 2 X and V be any regular open set of Y containing f(x). Since f is a.st.h.p.c., there exists U 2 PO(X, x) such that f(pcl(U)) V. Since X is submaximal, U is open and pcl(U) = cl(U). Therefore, we obtain f(cl(U)) V. This shows that f is a.st.h-c. h Corollary 3.11. Let X be a submaximal space and Y be a semi-regular space. Then the following properties are equivalent for f : X ! Y: (a) (b) (c) (d)
f f f f
is is is is
a.st.h.p.c.; st.h.p.c.; a.st.h-c.; strongly h-continuous.
Proof. It follows from Theorems 3.6 and 3.10 and Theorem 4.1 of [24].
h
4. Some properties A space X is said to be pre-regular [26] if for each preclosed set F and each point x 2 X F, there exist disjoint preopen sets U and V such that x 2 U and F V. Theorem 4.1. Let f : X ! Y be a function and g : X ! X · Y be the graph function of f. Then, the following properties hold: (a) If g is a.st.h.p.c., then f is a.st.h.p.c. and X is almost p-regular. (b) If f is a.st.h.p.c. and X is pre-regular, then g is a.st.h.p.c. Proof (a) Suppose that g is a.st.h.p.c. First, we show that f is a.st.h.p.c. Let x 2 X and V be a regular open set of Y containing f(x). Then X · V is a regular open set of X · Y containing g(x). Since g is a.st.h.p.c., there exists U 2 PO(X, x) such that g(pcl(U)) X · V. Therefore, we obtain f(pcl(U)) V. Next, we show that X is almost p-regular. Let U be any regular open set of X and x 2 U. Since g(x) 2 U · Y and U · Y is regular open in X · Y,
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there exists G 2 PO(X, x) such that g(pcl(G)) U · Y. Therefore, we obtain x 2 G pcl(G) U and hence X is almost p-regular. (b) Let x 2 X and W be any regular open set of X · Y containing g(x). There exist regular open sets U1 X and V Y such that g(x) = (x,f(x)) 2 U1 · V W. Since f is a.st.h.p.c., there exists U2 2 PO(X, x) such that f(pcl(U2)) V. Since X is pre-regular and U1 \ U2 2 PO(X, x), there exists U 2 PO(X, x) such that x 2 U pcl(U) U1 \ U2 [26, Lemma 4.2]. Therefore, we obtain g(pcl(U)) U1 · f(pcl(U2)) U1 · V W. This shows that g is a.st.h.p.c. h
Corollary 4.2. Let X be a pre-regular space. Then, a function f : X ! Y is a.st.h.p.c. if and only if the graph function g : X ! X · Y is a.st.h.p.c. Lemma 4.3 (Mashhour et al. [19]). Let A and X0 be subsets of a space X. (a) If A 2 PO(X) and X0 is semi-open in X, then A \ X0 2 PO(X). (b) If A 2 PO(X0) and X0 2 PO(X), then A 2 PO(X). Lemma 4.4 (Dontchev et al. [4]). Let A and X0 be subsets of a space X such that A X0 X. Let pclX 0 ðAÞ denote the preclosure of A in the subspace X0. (a) If X0 is semi-open in X, then pclX 0 ðAÞ pclðAÞ. (b) If A 2 PO(X0) and X0 2 PO(X), then pclðAÞ pclX 0 ðAÞ. Theorem 4.5. If f : X ! Y is a.st.h.p.c. and X0 is a semi-open subset of X, then the restriction fjX0 : X0 ! Y is a.st.h.p.c. Proof. For any x 2 X0 and any regular open set V of Y containing f(x), there exists U 2 PO(X, x) such that f(pcl(U)) V since f is a.st.h.p.c. Put U0 = U \ X0, then by Lemmas 4.3 and 4.4, U0 2 PO(X0,x) and pclX 0 (U0) pcl(U0). Therefore, we obtain ðf jX 0 ÞðpclX 0 ðU 0 ÞÞ ¼ f ðpclX 0 ðU 0 ÞÞ f ðpclðU 0 ÞÞ f ðpclðU ÞÞ V . This shows that fjX0 is a.st.h.p.c.
h
Theorem 4.6. A function f : X ! Y is a.st.h.p.c. if for each x 2 X there exists X0 2 PO(X, x) such that the restriction fjX0 : X0 ! Y is a.st.h.p.c. Proof. Let x 2 X and V be any regular open set of Y containing f(x). There exists X0 2 PO(X, x) such that fjX0 : X0 ! Y is a.st.h.p.c. Therefore, there exists U 2 PO(X0,x) such that ðf jX 0 ÞðpclX 0 ðU ÞÞ V . By Lemmas 4.3 and 4.4, U 2 PO(X, x) and pclðU Þ pclX 0 ðU Þ. Hence, we have, f ðpclðU ÞÞ ¼ ðf jX 0 ÞðpclðU ÞÞ ðF jX 0 ÞðpclX 0 ðU ÞÞ V . This shows that f is a.st.h.p.c. h In order to obtain some properties of the compositions of a.st.h.p.c. functions, we shall recall some definitions. Definition 4.7. A function f : X ! Y is said to be (a) pre-irresolute [30] if for each x 2 X and each V 2 PO(Y,f(x)), there exists U 2 PO(X, x) such that f(U) V, (b) M-pre-open [18] if f(U) 2 PO(Y) for each U 2 PO(X). Lemma 4.8 (Noiri [23]). If f : X ! Y is pre-irresolute and V is a pre-h-open set of Y, then f1(V) is pre-h-open in X. Theorem 4.9. Let f : X ! Y and g : Y ! Z be functions. Then, the following properties hold: (a) If f is a.st.h.p.c. and g is d-continuous, then the composition g f : X ! Z is a.st.h.p.c. (b) If f is pre-irresolute and g is a.st.h.p.c., then g f is a.st.h.p.c. (c) If f : X ! Y is an M-pre-open bijection and g f : X ! Z is a.st.h.p.c., then g is a.st.h.p.c.
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Proof (a) It is obvious from Theorem 3.1. (b) It follows immediately from Theorem 3.1 and Lemma 4.8. (c) Let W be any regular open set of Z. Since g f is a.st.h.p.c., (g f)1(W) is pre-h-open in X. Since f is M–pre-open and bijective, f1 is pre-irresolute and by Lemma 4.8, we have g1(W) = f((g f)1(W)) is pre-h-open in Y. Hence, by Theorem 3.1 g is a.st.h.p.c. h Let {Xa: a 2 K} be a family of spaces, Aa be a nonempty subset of Xa for each a 2 K and the product space {Xa: a 2 K} will be denoted by Xa. Q Q Lemma 4.10 (El-Deeb et al. [6]). Let n be a positive integer and A ¼ nj¼1 Aaj a6¼aj X a . (a) A 2 PO(X) if and only if Aaj 2 POðX aj Þ for each j = 1, . . . , n. (b) pcl(a2KAa) a2Kpcl(Aa).
Theorem 4.11. If a function fa : Xa ! Ya is a.st.h.p.c. for each a 2 K, the the product function f : Xa ! Ya, defined by f({xa}) = {fa(xa)} for each x = {xa}, is a.st.h.p.c. Proof. Let x = {xa} 2 Xa and W be any regular open set of Ya containing f(x). Then, there exists a regular open set V aj of Y aj such that n Y Y V aj Ya W . f ðxÞ ¼ ffa ðxa Þg 2 j¼1
a6¼aj
SinceQ fa is a.st.h.p.c. Q for each a, there exists U aj 2 POðX aj ; xaj Þ such that faj ðpclðU aj ÞÞ V aj for j = 1, . . . , n. Now, put U ¼ nj¼1 U aj a6¼aj X a . Then, by Lemma 4.10 we have U 2 PO(Xa,x) and ! n n n Y Y Y Y Y Y f ðpclðU ÞÞ f pclðU aj Þ Xa faj ðpclðU aj ÞÞ Ya V aj Ya W . a6¼aj
j¼1
This shows that f is a.st.h.p.c.
j¼1
a6¼aj
j¼1
a6¼aj
h
5. Separation axioms and a.st.h.p.c. functions Definition 5.1. A space X is said to be (a) pre-T2 (resp. pre-Urysohn) [25] if for each pair of distinct points x and y in X, there exist U 2 PO(X, x) and V 2 PO(X, y) such that U \ V = ; (resp. pcl(U) \ pcl(V) = ;); (b) rT0 [2] if for any two distinct points of X, there exists a regular open set containing one of the points but not the other. Theorem 5.2 (a) If f : X ! Y is an a.st.h.p.c. injection and Y is rT0, then X is pre-T2. (b) If f : X ! Y is an a.st.h.p.c. injection and Y is Hausdorff, then X is pre-Urysohn. Proof (a) Let x and y be any distinct points of X. Since f is injective, f(x) 5 f(y) and there exists either a regular open set V containing f(x) not containing f(y) or a regular open set W containing f(y) not containing f(x). If the first case holds, then there exists U 2 PO(X, x) such that f(pcl(U)) V. Therefore, we obtain f(y) 62 f(pcl(U)) and hence X pcl(U) 2 PO(X, y). If the second case holds, then we obtain a similar result. Therefore, X is pre-T2. (b) Let x and y be any distinct points of X. Then f(x) 5 f(y). Since Y is Hausdorff, there exist open sets V and W containing f(x) and f(y), respectively, such that int(cl(V)) \ int(cl(W)) = ;. Since f is a.st.h.p.c., there exist G 2 PO(X, x) and H 2 PO(X, y) such that f(pcl(G)) int(cl(V)) and f(pcl(H)) int(cl(W)). It follows that pcl(G) \ pcl(H) = ;. This shows that X is pre-Urysohn. h
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Corollary 5.3 (Noiri [23]). If f : X ! Y is a st.h.p.c. injection and Y is Hausdorff, then X is pre-Urysohn. Theorem 5.4. If f : X ! Y is an a.st.h.p.c. function and Y is Hausdorff, then the subset E = {(x Æ y) : f(x) = f(y)} is pre-hclosed in X · X. Proof. Suppose that (x,y) 62 E. Then f(x) 5 f(y). Since Y is Hausdorff, there exist open sets V and W of Y containing f(x) and f(y), respectively, such that int(cl(V)) \ int(cl(W)) = ;. Since f is a.st.h.p.c., there exist U 2 PO(X, x) and G 2 PO(X, y) such that f(pcl(U)) int(cl(V)) and f(pcl(G)) int(cl(W)). Set D = U · G. It follows that (x,y) 2 D 2 PO(X · X) and pcl(U · G) \ E [pcl(U) · pcl(G)] \ E = ;. Therefore, E is pre-h-closed in X · X. h Corollary 5.5 (Noiri [23]). If f : X ! Y is a st.h.p.c. function and Y is Hausdorff, then the subset E = {(x.y) : f(x) = f(y)} is pre-h-closed in X · X. Recall that for a function f : X ! Y, the subset {(x,f(x)) : x 2 X} of X · Y is called the graph of f is denoted by G(f). Definition 5.6. The graph G(f) of a function f : X ! Y is said to be strongly pre-closed [23] (resp. pre-h-closed) if for each (x,y) 2 (X · Y) G(f), there exist U 2 PO(X, x) and an open set V in Y containing y such that (pcl(U) · V) \ G(f) = ; (resp. (pcl(U) · cl(V)) \ G(f) = ;). Lemma 5.7. The graph G(f) of a function f : X ! Y is pre-h-closed if and only if for each (x,y) 2 (X · Y) G(f), there exist U 2 PO(X, x) and an open set V in Y containing y such that f(pcl(U)) \ cl(V) = ;. Theorem 5.8. If f : X ! Y is a.st.h.p.c. and Y is Hausdorff, then G(f) is pre-h-closed in X · Y. Proof. Let (x,y) 2 (X · Y) G(f). Then f(x) 5 y. Since Y is Hausdorff, there exist open sets V and W in Y containing f(x) and y, respectively, such that int(cl(V)) \ cl(W) = ;. Since f is a.st.h.p.c., there exists U 2 PO(X, x) such that f(pcl(U)) int(cl(V)). Therefore, f(pcl(U)) \ cl(W) = ; and then by Lemma 5.7 G(f) is pre-h-closed in X · Y. h Corollary 5.9 (Noiri [23]). If f : X ! Y is st.h.p.c. and Y is Hausdorff, then G(f) is strongly pre-closed in X · Y. 6. Covering properties Definition 6.1. A space X is said to be (a) (b) (c) (d) (e)
quasi-H-closed [28] if every cover of X by open sets has finite subcover whose closures cover of X; nearly compact [33] if every cover of X by regular open sets has a finite subcover; nearly countably compact [10] if every countable cover of X by regular open sets has a finite subcover; p-closed [4] if every cover of X by pre-open sets has a finite subcover whose preclosures cover X; countably p-closed [23] if every countable cover of X by pre-open sets has a finite subcover whose preclosures cover X.
A subset K of a space X is said to be quasi-H-closed relative to X [28] (resp. N-closed relative to X [33], p-closed relative to X [4]) if for every cover {Va:a 2 K} of K by open (resp. regular open, pre-open) sets of X, there exists a finite subset K0 of K such that K [{cl(Va):a 2 K0} (resp. K [{Va:a 2 K0}, K [{pcl(Va):a 2 K0}). Theorem 6.2. If f : X ! Y is an a.st.h.p.c. function and K is p-closed relative to X, then f(K) is N-closed relative to Y. Proof. Let {Va : a 2 K} be a cover of f(K) by regular open sets of Y. For each point x 2 K, there exists a(x) 2 K such that f(x) 2 Va(x). Since f is a.st.h.p.c., there exists Ux 2 PO(X, x) such that f(pcl(Ux)) Va(x). The family {Ux : x 2 K} is a cover of K by pre-open sets of X and hence there exists a finite subset K0 of K such that K [x2K 0 pclðU x Þ. Therefore, we obtain f ðKÞ [x2K 0 V aðxÞ . This shows that f(K) is N-closed relative to Y. h Corollary 6.3. Let f : X ! Y be an a.st.h.p.c. surjection. Then, the following properties hold: (a) If X is p-closed, then Y is nearly compact. (b) If X is countably p-closed, then Y is nearly countably compact.
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Theorem 6.4. If a function f : X ! Y has a pre-h-closed graph, then f(K) is h-closed in Y for each subset K which is pclosed relative to X. Proof. Let K be a p-closed relative to X and y 2 Y f(K). Then for each x 2 K we have (x,y) 62 G(f) and by Lemma 5.7 there exist Ux 2 PO(X, x) and an open set Vx of Y containing y such that f(pcl(Ux)) \ cl(Vx) = ;. The family {Ux : x 2 K} is a cover of K by pre-open sets of X. Since K is p-closed relative to X, there exists a finite subset K0 of K such that K [{pcl(Ux) : x 2 K0}. Put V = \{Vx : x 2 K0}. Then V is an open set containing y and f ðKÞ \ clðV Þ [x2K 0 f ðpclðU x ÞÞ \ clðV Þ [x2K 0 ½f ðpclðU x ÞÞ \ clðV x Þ ¼ ;. Therefore, we have y 2 h-cl(f(K)) and hence f(K) is h-closed in Y.
h
Theorem 6.5. Let X be a submaximal space. If a function f : X ! Y has a pre-h-closed graph, then f1(K) is h-closed in X for each subset K which is quasi-H-closed relative to Y. Proof. Let K be a quasi-H-closed set of Y and x 62 f1(K). Then for each y 2 K we have (x,y) 62 G(f) and by Lemma 5.7 there exists Uy 2 PO(X, x) and an open set Vy of Y containing y such that f(pcl(Uy)) \ cl(Vy) = ;. The family {Vy: y 2 K} is an open cover of K and there exists a finite subset K0 of K such that K [y2K 0 clðV y Þ. Since X is submaximal, each Uy is open in X and pcl(Uy) = cl(Uy). Set U ¼ \y2K 0 U y , then U is an open set containing x and f ðclðU ÞÞ \ clðKÞ [y2K 0 f ðclðU ÞÞ \ clðV y Þ [x2K 0 f ðpclðU y ÞÞ \ clðV y Þ ¼ ;. Therefore, we have cl(U) \ f1(K) = ; and hence x 62 h-cl(f1(K)). This shows that f1(K) is h-closed in X.
h
Corollary 6.6. Let X be a submaximal space and Y be a quasi-H-closed Hausdorff space. The following properties are equivalent for a function f : X ! Y: (a) f is a.st.h.p.c.; (b) G(f) is pre-h-closed in X · Y; (c) f is a.st.h-c. Proof (a) ) (b): This follows from Theorem 5.8. (b) ) (c): Let F be any regular closed set Y. By (2.2) of [28] F is quasi-H-closed and hence quasi-H-closed relative to Y. It follows from Theorem 6.5 that f1(F) is h-closed in X. Therefore, f is a.st.h-c. (c) ) (a): Obvious. h In conclusion, we study almost strongly h-precontinuity on topological spaces and obtain several improvements of results established by Noiri [23]. The generalizations of continuity related to some topological properties such as separation axioms, compactness, connectedness will been found to be useful in the study of topology. Thus we may stress the importance of almost strongly h-precontinuity as a branch of them and the possible application in digital topology, quantum physics, high energy physics and superstring theory.
References [1] Abd El-Monsef ME, El-Deeb SN, Mahmoud RA. b-open sets and b-continuous mappings. Bull Fac Sci Assiut Univ 1983; 12(1):77–90. [2] Andrijevic´ D. Semi-preopen sets. Mat Vesnik 1986;38:24–32. [3] Andrijevic´ D. On b-open sets. Mat Vesnik 1996;48:59–64. [4] Dontchev J, Ganster M, Noiri T. On p-closed spaces. Int J Math Math Sci 2000;24:203–12. [5] Dontchev J, Przemski M. On the various decompositions of continuous and some weakly continuous functions. Acta Math Hungar 1996;71(1–2):109–20. [6] El-Deeb N, Hasanein IA, Mashhour AS, Noiri T. On p-regular spaces. Bull Math Soc Sci Math R S Roumanie 1983;27(75):311–5. [7] El Naschie MS, Rossler OE, Oed G. Information and diffusion in quantum physics. Chaos, Solitons & Fractals 1996;7(5). Special Issue. [8] El Naschie MS. ‘‘Quantum gravity’’, Clifford algebra, Fuzzy set theory and the fundamental constants of nature. Chaos, Solitons & Fractals 2004;20:437–50.
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[9] El Naschie MS. The symplictic Vacuum, Exotic quasi-particles and gravitational instanton. Chaos, Solitons & Fractals 2004;22: 1–11. [10] Ergun N. On nearly paracompact spaces. Istanbul Univ Fen Mec Ser A 1980;45:65–87. [11] Frolı´k Z. Remarks concerning the invariance of Baire spaces under mappings. Czechoslovak Math J 1961;11(86):381–5. [12] Husain T. Almost continuous mappings. Prace Mat 1966;10:1–7. [13] Jankovic´ DS. h-regular spaces. Int J Math Math Sci 1985;8:615–9. [14] Levine N. Semi-open sets and semi-continuity in topological spaces. Amer Math Monthly 1963;70:36–41. [15] Long PE, Herrington LL. Strongly h-continuous functions. J Korean Math Soc 1981;18:21–8. [16] Malghan SR, Navalagi GB. Almost p-regular, p-completely regular and almost p-completely regular spaces. Bull Math 1990; 80(4):317–26. [17] Mashhour AS, Abd El-Monsef ME, El-Deeb SN. On precontinuous and weak precontinuous functions. Proc Math Phys Soc Egypt 1982;53:47–53. [18] Mashhour AS, Abd El-Monsef ME, Hasanein IA. On pretopological spaces. Bull Math Soc Sci Math RS Roumanie 1984; 28(76):39–45. [19] Mashhour AS, Hasanein IA, El-Deeb SN. A note on semi-continuity and precontinuity. Indian J Pure Appl Math 1982;13: 1119–23. [20] Nasef AA, Noiri T. Some weak forms of almost continuity. Acta Math Hungar 1997;74(3):211–9. [21] Nja˚stad O. On some classes of nearly open sets. Pacific J Math 1965;15:961–70. [22] Noiri T. On d-continuous functions. J Korean Math Soc 1980;16(2):161–6. [23] Noiri T. Strongly h-precontinuous functions. Acta Math Hungar 2001;90(4):307–16. [24] Noiri T, Kang SM. On almost strongly h-continuous functions. Indian J Pure Appl Math 1984;15(1):1–8. [25] Nour TMJ. Contributions to the Theory of Bitopological Spaces. PhD Thesis. Univ of Delhi, 1989. [26] Pal MC, Bhattacharyya P. Feeble and strong forms of preirresolute functions. Bull Malaysian Math Soc 1996;19:63–75. [27] Popa V, Noiri T. Almost weakly continuous functions. Demonstratio Math 1992;25:241–51. [28] Porter J, Thomas J. On H-closed and minimal Hausdorff spaces. Trans Am Math Soc 1974;138:159–70. [29] Pta´k V. Completeness and open mapping theorems. Bull Soc Math France 1958;86:41–74. [30] Reilly IL, Vamanamurthy MK. On a-continuity in topological spaces. Acta Math Hungar 1985;45:27–32. [31] Reilly IL, Vamanamurthy MK. On some questions concerning preopen sets. Kyungpook Math J 1990;30:87–93. [32] Singal MK, Arya SP. On almost-regular space. Glasnik Mat III 1969;4(24):89–99. [33] Singal MK, Mathur A. On nearly-compact spaces. Boll UMI 1969;2(4):702–10. [34] Singal MK, Singal AR. Almost-continuous mappings. Yokohama Math J 1968;16:63–73. [35] Svozil K. Quantum field theory on fractal space-time: A new regularization method. J Phy A Math Gen 1987;20:3861–75. [36] Velic˘ko NV. H-closed topological spaces. Amer Math Soc Transl 1968;78:103–18.
Almost strongly h-precontinuous functions
q
Jin Han Park *, Sang Wook Bae, Yong Beom Park Division of Mathematical Sciences, Pukyong National University, 599-1 Daeyeon 3-Dong, Nam-Gu, Pusan 608-737, South Korea Accepted 6 May 2005
Abstract In this paper, we introduce a new class of functions called almost strongly h-precontinuous function which is a generalization of almost strongly h-continuous functions due to Noiri and Kang [Noiri T, Kang SM. On almost strongly hcontinuous functions. Indian J Pure Appl Math 1984;15(1):1–8] and strongly h-precontinuous functions due to Noiri [Noiri T. Strongly h-precontinuous functions. Acta Math Hung 2001;90(4):307–16]. Some characterizations and several properties concerning almost strongly h-precontinuous functions are obtained. The relationships between almost strongly h-precontinuity and other types of continuity are also given. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Relation of topology and physics have been appeared in [7–9,35], i.e. topology plays a significant role in quantum physics, high energy physics and superstring theory. Continuity on topological spaces, as important and basic subject in study of topology, have been researched by many mathematicians. This concept has been generalized by weaker forms of open sets such as a-open sets [21], semi-open sets [14], pre-open sets [17], b-open sets [3] and b-open sets [1]. In 1984, Mashhour et al. [17] introduced and investigated the notion of precontinuous functions. Precontinuity was called near continuity by Pta´k [29] and also almost continuity by Frolı´k [11] and Husain [12]. Jankovic´ [13] introduced almost weak continuity as a weak form of precontinuity. Popa and Noiri [27] introduced weak precontinuity and showed that almost weak continuity is equivalent to weak precontinuity. Recently, Noiri [23] introduced and investigated the notion of strongly h-precontinuous functions which is implied by that of strongly h-continuous functions [15] and implies that of precontinuous functions. In this paper, we introduce a new class of functions called almost strongly h-precontinuous functions which is contained in the class of weakly precontinuous functions and contains both the class of almost strongly h-continuous functions [24] and the class of strongly h-precontinuous functions. We investigate almost strongly h-precontinuous functions and obtain several improvements of results established by Noiri [23]. It is also shown that every almost strongly h-precontinuous surjective image of p-closed (resp. countably p-closed) space is nearly compact (resp. nearly countably compact).
q *
This work was supported by grant no. R05-2002-000-01045-0 from the Korea Science & Engineering Foundation. Corresponding author. E-mail address: [email protected] (J.H. Park).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.05.058
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2. Preliminaries Throughout this paper, spaces (X, s) and (Y, r) (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. For a subset A of (X, s), cl(A) and int(A) represent the closure of A and the interior of A with respect to s, respectively. A subset A is said to be regular open (resp. regular closed) if A = int(cl(A)) (resp. A = cl(int(A))). A point x of X is called a h-cluster [36] (resp. d-cluster [36]) point of A if cl(U) \ A 5 ; (resp. int(cl(U)) \ A 5 ;) for every open set U of X containing x. The set of all h-cluster (resp. d-cluster) points of A is called the h-closure [36] (resp. d-closure [36]) of A and is denoted by h-cl(A) (resp. dcl(A)). A subset A is said to be h-closed [36] (resp. d-closed [36]) if h-cl(A) = A (resp. d-cl(A) = A). The complement of a h-closed (resp. d-closed) set is said to be h-open (resp. d-open). A subset A is said to be pre-open [17] (resp. a-open [21], semi-open [14], b-open [1]) if A int(cl(A)) (resp. A int(cl(int(A))), A cl(int(A)), A cl(int(cl(A)))). The complement of a pre-open (resp. a-open, semi-open, b-open) set is said to be preclosed (resp. a-closed, semi-closed, b-closed). The family of all pre-open sets of X is denoted by PO(X) and the family {U 2 PO(X) : x 2 U} is denoted by PO(X, x), where x is a point of X. The intersection of all preclosed sets of X containing A is called the preclosure [6] of A and is denoted by pcl(A). The a-closure, semi-closure and b-closure are similarly defined and are denoted by a-cl(A), scl(A) and b-cl(A) [5,20]. The union of all pre-open sets of X contained in A is called preinterior and is denoted by pint(A). A point x of X is called a pre-h-cluster point of A if pcl(U) \ A 5 ; for every pre-open set U of X containing x. The set of all pre-h-cluster points of A is called the pre-hclosure of A and is denoted by h-pcl(A). A subset A is said to be pre-h-closed [26] if A = h-pcl(A). The complement of a pre-h-closed set is said to be pre-h-open. Definition 2.1. A function f : X ! Y is said to be (a) almost continuous [34] (briefly, a.c.S.) if for each x 2 X and each open set V of Y containing f(x), there exists an open set U containing x such that f(U) int(cl(V)); (b) d-continuous [22] if for each x 2 X and each open set V of Y containing f(x), there exists an open set U containing x such that f(int(cl(U))) int(cl(V)); (c) precontinuous [17] or almost continuous [12] if for each x 2 X and each open set V of Y containing f(x), there exists U 2 PO(X, x) such that f(U) V; (d) weakly precontinuous [27] or almost weakly continuous [13] if for each x 2 X and each open set V of Y containing f(x), there exists U 2 PO(X, x) such that f(U) cl(V); (e) strongly h-precontinuous [23] (briefly, st.h.p.c.) if for each x 2 X and each open set V of Y containing f(x), there exists U 2 PO(X, x) such that f(pcl(U)) V. Definition 2.2. A function f : X ! Y is said to be almost strongly h-precontinuous (briefly, a.st.h.p.c.) if for each x 2 X and each open set V of Y containing f(x), there exists U 2 PO(X, x) such that f(pcl(U)) int(cl(V)). Definition 2.3. A function f : X ! Y is said to be strongly h-continuous [22] (resp. almost strongly h-continuous [24] (briefly, a.st.h-c.)) if for each x 2 X and each open set V of Y containing f(x), there exists an open neighborhood U of x such that f(cl(U)) V (resp. f(cl(U)) int(cl(V))). Remark 2.4. Almost strongly h-precontinuity is implied by both almost strongly h-continuity and strongly h-precontinuity and implies weak precontinuity. None of these implications is reversible as the following examples show. Moreover, the following Examples 2.5 and 2.7 show that almost strongly h-precontinuity and continuity are independent of each other. Example 2.5. Let X = {a, b, c, d}, s = {X, ;, {c}, {a, b}, {a, b, c}} and r = {X, ;, {a}, {c}, {a, b}, {a, c}, {a, b, c}, {a, c, d}}. Define a function f : (X, s) ! (X, r) as follows: f(a) = f(b) = b and f(c) = f(d) = a. Then f is a.st.h.p.c. (even a.st.h-c.) but it is neither continuous nor st.h.p.c. Example 2.6. Let (X, s) and (X, r) be the spaces in Example 2.5. Define a function f : (X,s) ! (X,r) as follows: f(a) = b and f(b) = f(c) = f(d) = c. Then f is a.st.h.p.c. but it is not a.st.h-c. Example 2.7. Let (X, s) be the space in Example 2.5. Then the identity function f : (X, s) ! (X, s) is continuous (hence weakly precontinuous) but not a.st.h.p.c.
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3. Characterizations Theorem 3.1. For a function f : X ! Y, the following are equivalent: (a) (b) (c) (d) (e) (f) (g) (h)
f is a.st.h.p.c.; f1(V) is pre-h-open in X for each regular open set V of Y; f1(F) is pre-h-closed in X for each regular closed set F of Y; for each x 2 X and each regular open set V of Y containing f(x), there exists U 2 PO(X, x) such that f(pcl(U)) V; f1(V) is pre-h-open in X for each d-open set V of Y; f1(F) is pre-h-closed in X for each d-closed set F of Y; f(h-pcl(A)) d-cl(f(A)) for each subset A of X; h-pcl(f1(B)) f1(d-cl(B)) for each subset B of Y.
Proof (a) ) (b): Let V be any regular open set of Y and x 2 f1(V). There exists U 2 PO(X, x) such that f(pcl(U)) V. Therefore, we have x 2 U pcl(U) f1(V). This shows that f1(V) is pre-h-open in X. (b) ) (c): Let F be any regular closed set of Y. By (b), f1(F) = X f1(Y F) is pre-h-closed in X. (c) ) (d): Let x 2 X and V be any regular open set of Y containing f(x). By (c), f1(Y V) = X f1(V) is pre-hclosed in X. Since f1(V) is a pre-h-open set containing x, there exists U 2 PO(X, x) such that pcl(U) f1(V). Therefore, we have f(pcl(U)) V. (d) ) (e): Let V be any d-open set of Y and x 2 f1(V). There exists a regular open set G of Y such that f(x) 2 G V. By (d), there exists U 2 PO(X, x) such that f(pcl(U)) G. Therefore, we obtain x 2 U pcl(U) f1(V). This shows that f1(V) is pre-h-open in X. (e) ) (f): Let F be any d-closed set of Y. By (e), f1(F) = X f1(Y F) is pre-h-closed in X. (f) ) (g): Let A be any subset of Y. Since d-cl(f(A)) is d-closed in Y, f1(d-cl(f(A))) is pre-h-closed in X. Let x 62 f1(d-cl(f(A))). There exists U 2 PO(X, x) such that pcl(U) \ f1(d-cl(f(A))) = ; and thus pcl(U) \ A = ;. Hence x 62 h-pcl(A). Therefore, we have f(h-pcl(A)) d-cl(f(A)). (g) ) (h): Let B be any subset of Y. By (g), we have f(h-pcl(f1(B))) d-cl(B) and hence h-pcl(f1(B)) f1(d-cl(B)). (h) ) (a): Let x 2 X and V be any open set of Y containing f(x). Then G = Y int(cl(V)) is regular closed and hence d-closed in Y. By (h), h-pcl(f1(G)) f1(G) and hence f1(G) is pre-h-closed in X. Therefore, f1(int(cl(V))) is pre-hopen set containing x. There exists U 2 PO(X, x) such that pcl(U) f1(int(cl(V))). Therefore, we obtain f(pcl(U)) int(cl(V)). This shows that f is a.st.h.p.c. h It is known that the family of all d-open sets in a space (X,s) form a topology for X which is denoted by sd. However, sd is identical with the semiregularization ss of s and hence we use ss in the place of sd. For simplicity, we shall denote (X,ss) by Xs. Lemma 3.2 (Andrijevic´ [2]). scl(V) = int(cl(V)) for each pre-open set V of a space X. Theorem 3.3. For a function f : X ! Y the following are equivalent: (a) (b) (c) (d)
f is a.st.h.p.c.; for each x 2 X and each open set V of Y containing f(x), there exists U 2 PO(X, x) such that f(pcl(U)) scl(V); f1(V) h-pcl(f1(int(cl(V)))) for each open set V of Y; f : X ! Ys is st.h.p.c.
Proof (a) () (b): It follows from Lemma 3.2. (a) ) (c): Let V be any open set of Y and x 2 f1(V). By (a), there exists U 2 PO(X, x) such that f(pcl(U)) int(cl(V)). Therefore, we have x 2 U pcl(U) f1(int(cl(V))) and hence x 2 h-pcl(f1(int(cl(V)))). It follows that f1(V) h-pcl(f1(int(cl(V)))). (c) ) (d): Let x 2 X and V be any open set of Ys containing f(x). There exists a regular open set G of Y such that f(x) 2 G V. By (c), we have x 2 f1(G) h-pcl(f1(G)) and hence there exists U 2 PO(X) such that x 2 U pcl(U) f1(G). Therefore, we obtain f(pcl(U)) V. This shows that f : X ! Ys is st.h.p.c.
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(d) ) (a): Let V be regular open set of Y. For any x 2 f1(V), f(x) V and V is open in Ys. There exists U 2 PO(X, x) such that f(pcl(U)) V and hence pcl(U) f1(V). Therefore, we have f1(V) h-pcl(f1(V)) and f1(V) is pre-h-open in X. By Theorem 3.1, f is a.st.h.p.c. h
Lemma 3.4 [24]. For a subset V of a space X, then following hold: (a) a-cl(V) = cl(V) for each b-open set V of X. (b) pcl(V) = cl(V) for each semi-open set V of X.
Theorem 3.5. For a function f : X ! Y, the following are equivalent: (a) (b) (c) (d) (e)
f is a.st.h.p.c.; h-pcl(f1(V)) f1(cl(V)) for each b-open set V of Y; h-pcl(f1(V)) f1(cl(V)) for each semi-open set V of Y; h-pcl(f1(V)) f1(a-cl(V)) for each b-open set V of Y; h-pcl(f1(V)) f1(pcl(V)) for each semi-open set V of Y.
Proof (a) ) (b): Let V be any b-open set of Y. Then by Theorem 2.4 in [2], cl(V) is regular closed in Y. Since f is a.st.h.p.c., f1(cl(V)) is pre-h-closed in X and hence h-pcl(f1(V)) f1(cl(V)). (b) ) (c): This is obvious since every semi-open set is b-open. (c) ) (a): Let F be any regular closed set of Y. Then F is semi-open in Y and hence hpcl(f1(F)) f1(cl(F)) = f1(F). This shows that f1(F) is pre-h-closed in X. Therefore, f is a.st.h.p.c. (b) () (d): It follows from Lemma 3.4(a). (c) () (e): It follows from Lemma 3.4(b). h Recall that a space X is said to be almost regular [32] (resp. semi-regular) if for any regular open (resp. open) set U of X and each point x 2 U, there exits a regular open set V of X such that x 2 V cl(V) U (resp. x 2 V U). Theorem 3.6. Let f : X ! Y be a function. Then, the following properties hold: (a) If f is precontinuous and Y is almost regular, then f is a.st.h.p.c. (b) If f is a.st.h.p.c. and Y is semi-regular, then f is st.h.p.c.
Proof (a) Let x 2 X and V be any regular open set of Y containing f(x). Since Y is almost regular, there exists an open set W such that f(x) 2 W cl(W) V. Since f is precontinuous, there exists U 2 PO(X, x) such that f(U) W. We shall show that f(pcl(U)) cl(W). Suppose that y 62 cl(W). There exists an open neighborhood G of y such that G \ W = ;. Since f is precontinuous, f1(G) 2 PO(X) and f1(G) \ U = ; and hence f1(G) \ pcl(U) = ;. Therefore, we obtain G \ f(pcl(U)) = ; and y 62 f(pcl(U)). Consequently, we have f(pcl(U)) cl(W) V. (b) Let x 2 X and V be any open set of Y containing f(x). Since Y is semi-regular, there exists a regular open set W such that f(x) W V. Since f is a.st.h.p.c., there exists U 2 PO(X, x) such that f(pcl(U)) W. Therefore, we have f(pcl(U)) V. h Corollary 3.7. Let Y be a regular space. Then, the following properties are equivalent for a function f : X ! Y: (a) (b) (c) (d)
f f f f
is is is is
weakly precontinuous; precontinuous; a.st.h.p.c.; st.h.p.c.
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Proof. It follows from Theorem 3.2 of [23] and Theorem 3.6.
h
Definition 3.8. A space X is said to be p-regular [6] (resp. almost p-regular [16]) if for each closed (resp. regular closed) set F and each point x 2 X F, there exist disjoint pre-open sets U and V such that x 2 U and F V. Theorem 3.9 (a) If continuous function f : X ! Y is a.st.h.p.c., then X is almost p-regular. (b) If f : X ! Y is a.c.S. (resp. d-continuous) and X is p-regular (resp. almost p-regular), then f is a.st.h.p.c. Proof (a) Let f : X ! X be the identity. Then f is continuous and hence a.st.h.p.c. For any regular open set U of X and any point x of U, we have f(x) = x 2 U and there exists G 2 PO(X, x) such that f(pcl(G)) U. Therefore, we have x 2 G pcl(G) U and hence X is almost p-regular. (b) Suppose that f : X ! Y is a.c.S. (resp. d-continuous) and X is p-regular (resp. almost p-regular). For each x 2 X and any regular open set V containing f(x), f1(V) is an open (resp. regular open) set of X containing x. Since X is p-regular (resp. almost p-regular), there exists U 2 PO(X, x) such that x 2 U pcl(U) f1(V). Therefore, we have f(pcl(U)) V. This shows that f is a.st.h.p.c. h A space X is said to be submaximal [31] if each dense subset of X is open in X. It is shown in [31] that a space X is submaximal if and only if every pre-open set of X is open. Theorem 3.10. Let X be a submaximal space. Then f : X ! Y is a.st.h.p.c. if and only if f is a.st.h-c. Proof. Suppose that f is a.st.h.p.c. Let x 2 X and V be any regular open set of Y containing f(x). Since f is a.st.h.p.c., there exists U 2 PO(X, x) such that f(pcl(U)) V. Since X is submaximal, U is open and pcl(U) = cl(U). Therefore, we obtain f(cl(U)) V. This shows that f is a.st.h-c. h Corollary 3.11. Let X be a submaximal space and Y be a semi-regular space. Then the following properties are equivalent for f : X ! Y: (a) (b) (c) (d)
f f f f
is is is is
a.st.h.p.c.; st.h.p.c.; a.st.h-c.; strongly h-continuous.
Proof. It follows from Theorems 3.6 and 3.10 and Theorem 4.1 of [24].
h
4. Some properties A space X is said to be pre-regular [26] if for each preclosed set F and each point x 2 X F, there exist disjoint preopen sets U and V such that x 2 U and F V. Theorem 4.1. Let f : X ! Y be a function and g : X ! X · Y be the graph function of f. Then, the following properties hold: (a) If g is a.st.h.p.c., then f is a.st.h.p.c. and X is almost p-regular. (b) If f is a.st.h.p.c. and X is pre-regular, then g is a.st.h.p.c. Proof (a) Suppose that g is a.st.h.p.c. First, we show that f is a.st.h.p.c. Let x 2 X and V be a regular open set of Y containing f(x). Then X · V is a regular open set of X · Y containing g(x). Since g is a.st.h.p.c., there exists U 2 PO(X, x) such that g(pcl(U)) X · V. Therefore, we obtain f(pcl(U)) V. Next, we show that X is almost p-regular. Let U be any regular open set of X and x 2 U. Since g(x) 2 U · Y and U · Y is regular open in X · Y,
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there exists G 2 PO(X, x) such that g(pcl(G)) U · Y. Therefore, we obtain x 2 G pcl(G) U and hence X is almost p-regular. (b) Let x 2 X and W be any regular open set of X · Y containing g(x). There exist regular open sets U1 X and V Y such that g(x) = (x,f(x)) 2 U1 · V W. Since f is a.st.h.p.c., there exists U2 2 PO(X, x) such that f(pcl(U2)) V. Since X is pre-regular and U1 \ U2 2 PO(X, x), there exists U 2 PO(X, x) such that x 2 U pcl(U) U1 \ U2 [26, Lemma 4.2]. Therefore, we obtain g(pcl(U)) U1 · f(pcl(U2)) U1 · V W. This shows that g is a.st.h.p.c. h
Corollary 4.2. Let X be a pre-regular space. Then, a function f : X ! Y is a.st.h.p.c. if and only if the graph function g : X ! X · Y is a.st.h.p.c. Lemma 4.3 (Mashhour et al. [19]). Let A and X0 be subsets of a space X. (a) If A 2 PO(X) and X0 is semi-open in X, then A \ X0 2 PO(X). (b) If A 2 PO(X0) and X0 2 PO(X), then A 2 PO(X). Lemma 4.4 (Dontchev et al. [4]). Let A and X0 be subsets of a space X such that A X0 X. Let pclX 0 ðAÞ denote the preclosure of A in the subspace X0. (a) If X0 is semi-open in X, then pclX 0 ðAÞ pclðAÞ. (b) If A 2 PO(X0) and X0 2 PO(X), then pclðAÞ pclX 0 ðAÞ. Theorem 4.5. If f : X ! Y is a.st.h.p.c. and X0 is a semi-open subset of X, then the restriction fjX0 : X0 ! Y is a.st.h.p.c. Proof. For any x 2 X0 and any regular open set V of Y containing f(x), there exists U 2 PO(X, x) such that f(pcl(U)) V since f is a.st.h.p.c. Put U0 = U \ X0, then by Lemmas 4.3 and 4.4, U0 2 PO(X0,x) and pclX 0 (U0) pcl(U0). Therefore, we obtain ðf jX 0 ÞðpclX 0 ðU 0 ÞÞ ¼ f ðpclX 0 ðU 0 ÞÞ f ðpclðU 0 ÞÞ f ðpclðU ÞÞ V . This shows that fjX0 is a.st.h.p.c.
h
Theorem 4.6. A function f : X ! Y is a.st.h.p.c. if for each x 2 X there exists X0 2 PO(X, x) such that the restriction fjX0 : X0 ! Y is a.st.h.p.c. Proof. Let x 2 X and V be any regular open set of Y containing f(x). There exists X0 2 PO(X, x) such that fjX0 : X0 ! Y is a.st.h.p.c. Therefore, there exists U 2 PO(X0,x) such that ðf jX 0 ÞðpclX 0 ðU ÞÞ V . By Lemmas 4.3 and 4.4, U 2 PO(X, x) and pclðU Þ pclX 0 ðU Þ. Hence, we have, f ðpclðU ÞÞ ¼ ðf jX 0 ÞðpclðU ÞÞ ðF jX 0 ÞðpclX 0 ðU ÞÞ V . This shows that f is a.st.h.p.c. h In order to obtain some properties of the compositions of a.st.h.p.c. functions, we shall recall some definitions. Definition 4.7. A function f : X ! Y is said to be (a) pre-irresolute [30] if for each x 2 X and each V 2 PO(Y,f(x)), there exists U 2 PO(X, x) such that f(U) V, (b) M-pre-open [18] if f(U) 2 PO(Y) for each U 2 PO(X). Lemma 4.8 (Noiri [23]). If f : X ! Y is pre-irresolute and V is a pre-h-open set of Y, then f1(V) is pre-h-open in X. Theorem 4.9. Let f : X ! Y and g : Y ! Z be functions. Then, the following properties hold: (a) If f is a.st.h.p.c. and g is d-continuous, then the composition g f : X ! Z is a.st.h.p.c. (b) If f is pre-irresolute and g is a.st.h.p.c., then g f is a.st.h.p.c. (c) If f : X ! Y is an M-pre-open bijection and g f : X ! Z is a.st.h.p.c., then g is a.st.h.p.c.
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Proof (a) It is obvious from Theorem 3.1. (b) It follows immediately from Theorem 3.1 and Lemma 4.8. (c) Let W be any regular open set of Z. Since g f is a.st.h.p.c., (g f)1(W) is pre-h-open in X. Since f is M–pre-open and bijective, f1 is pre-irresolute and by Lemma 4.8, we have g1(W) = f((g f)1(W)) is pre-h-open in Y. Hence, by Theorem 3.1 g is a.st.h.p.c. h Let {Xa: a 2 K} be a family of spaces, Aa be a nonempty subset of Xa for each a 2 K and the product space {Xa: a 2 K} will be denoted by Xa. Q Q Lemma 4.10 (El-Deeb et al. [6]). Let n be a positive integer and A ¼ nj¼1 Aaj a6¼aj X a . (a) A 2 PO(X) if and only if Aaj 2 POðX aj Þ for each j = 1, . . . , n. (b) pcl(a2KAa) a2Kpcl(Aa).
Theorem 4.11. If a function fa : Xa ! Ya is a.st.h.p.c. for each a 2 K, the the product function f : Xa ! Ya, defined by f({xa}) = {fa(xa)} for each x = {xa}, is a.st.h.p.c. Proof. Let x = {xa} 2 Xa and W be any regular open set of Ya containing f(x). Then, there exists a regular open set V aj of Y aj such that n Y Y V aj Ya W . f ðxÞ ¼ ffa ðxa Þg 2 j¼1
a6¼aj
SinceQ fa is a.st.h.p.c. Q for each a, there exists U aj 2 POðX aj ; xaj Þ such that faj ðpclðU aj ÞÞ V aj for j = 1, . . . , n. Now, put U ¼ nj¼1 U aj a6¼aj X a . Then, by Lemma 4.10 we have U 2 PO(Xa,x) and ! n n n Y Y Y Y Y Y f ðpclðU ÞÞ f pclðU aj Þ Xa faj ðpclðU aj ÞÞ Ya V aj Ya W . a6¼aj
j¼1
This shows that f is a.st.h.p.c.
j¼1
a6¼aj
j¼1
a6¼aj
h
5. Separation axioms and a.st.h.p.c. functions Definition 5.1. A space X is said to be (a) pre-T2 (resp. pre-Urysohn) [25] if for each pair of distinct points x and y in X, there exist U 2 PO(X, x) and V 2 PO(X, y) such that U \ V = ; (resp. pcl(U) \ pcl(V) = ;); (b) rT0 [2] if for any two distinct points of X, there exists a regular open set containing one of the points but not the other. Theorem 5.2 (a) If f : X ! Y is an a.st.h.p.c. injection and Y is rT0, then X is pre-T2. (b) If f : X ! Y is an a.st.h.p.c. injection and Y is Hausdorff, then X is pre-Urysohn. Proof (a) Let x and y be any distinct points of X. Since f is injective, f(x) 5 f(y) and there exists either a regular open set V containing f(x) not containing f(y) or a regular open set W containing f(y) not containing f(x). If the first case holds, then there exists U 2 PO(X, x) such that f(pcl(U)) V. Therefore, we obtain f(y) 62 f(pcl(U)) and hence X pcl(U) 2 PO(X, y). If the second case holds, then we obtain a similar result. Therefore, X is pre-T2. (b) Let x and y be any distinct points of X. Then f(x) 5 f(y). Since Y is Hausdorff, there exist open sets V and W containing f(x) and f(y), respectively, such that int(cl(V)) \ int(cl(W)) = ;. Since f is a.st.h.p.c., there exist G 2 PO(X, x) and H 2 PO(X, y) such that f(pcl(G)) int(cl(V)) and f(pcl(H)) int(cl(W)). It follows that pcl(G) \ pcl(H) = ;. This shows that X is pre-Urysohn. h
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Corollary 5.3 (Noiri [23]). If f : X ! Y is a st.h.p.c. injection and Y is Hausdorff, then X is pre-Urysohn. Theorem 5.4. If f : X ! Y is an a.st.h.p.c. function and Y is Hausdorff, then the subset E = {(x Æ y) : f(x) = f(y)} is pre-hclosed in X · X. Proof. Suppose that (x,y) 62 E. Then f(x) 5 f(y). Since Y is Hausdorff, there exist open sets V and W of Y containing f(x) and f(y), respectively, such that int(cl(V)) \ int(cl(W)) = ;. Since f is a.st.h.p.c., there exist U 2 PO(X, x) and G 2 PO(X, y) such that f(pcl(U)) int(cl(V)) and f(pcl(G)) int(cl(W)). Set D = U · G. It follows that (x,y) 2 D 2 PO(X · X) and pcl(U · G) \ E [pcl(U) · pcl(G)] \ E = ;. Therefore, E is pre-h-closed in X · X. h Corollary 5.5 (Noiri [23]). If f : X ! Y is a st.h.p.c. function and Y is Hausdorff, then the subset E = {(x.y) : f(x) = f(y)} is pre-h-closed in X · X. Recall that for a function f : X ! Y, the subset {(x,f(x)) : x 2 X} of X · Y is called the graph of f is denoted by G(f). Definition 5.6. The graph G(f) of a function f : X ! Y is said to be strongly pre-closed [23] (resp. pre-h-closed) if for each (x,y) 2 (X · Y) G(f), there exist U 2 PO(X, x) and an open set V in Y containing y such that (pcl(U) · V) \ G(f) = ; (resp. (pcl(U) · cl(V)) \ G(f) = ;). Lemma 5.7. The graph G(f) of a function f : X ! Y is pre-h-closed if and only if for each (x,y) 2 (X · Y) G(f), there exist U 2 PO(X, x) and an open set V in Y containing y such that f(pcl(U)) \ cl(V) = ;. Theorem 5.8. If f : X ! Y is a.st.h.p.c. and Y is Hausdorff, then G(f) is pre-h-closed in X · Y. Proof. Let (x,y) 2 (X · Y) G(f). Then f(x) 5 y. Since Y is Hausdorff, there exist open sets V and W in Y containing f(x) and y, respectively, such that int(cl(V)) \ cl(W) = ;. Since f is a.st.h.p.c., there exists U 2 PO(X, x) such that f(pcl(U)) int(cl(V)). Therefore, f(pcl(U)) \ cl(W) = ; and then by Lemma 5.7 G(f) is pre-h-closed in X · Y. h Corollary 5.9 (Noiri [23]). If f : X ! Y is st.h.p.c. and Y is Hausdorff, then G(f) is strongly pre-closed in X · Y. 6. Covering properties Definition 6.1. A space X is said to be (a) (b) (c) (d) (e)
quasi-H-closed [28] if every cover of X by open sets has finite subcover whose closures cover of X; nearly compact [33] if every cover of X by regular open sets has a finite subcover; nearly countably compact [10] if every countable cover of X by regular open sets has a finite subcover; p-closed [4] if every cover of X by pre-open sets has a finite subcover whose preclosures cover X; countably p-closed [23] if every countable cover of X by pre-open sets has a finite subcover whose preclosures cover X.
A subset K of a space X is said to be quasi-H-closed relative to X [28] (resp. N-closed relative to X [33], p-closed relative to X [4]) if for every cover {Va:a 2 K} of K by open (resp. regular open, pre-open) sets of X, there exists a finite subset K0 of K such that K [{cl(Va):a 2 K0} (resp. K [{Va:a 2 K0}, K [{pcl(Va):a 2 K0}). Theorem 6.2. If f : X ! Y is an a.st.h.p.c. function and K is p-closed relative to X, then f(K) is N-closed relative to Y. Proof. Let {Va : a 2 K} be a cover of f(K) by regular open sets of Y. For each point x 2 K, there exists a(x) 2 K such that f(x) 2 Va(x). Since f is a.st.h.p.c., there exists Ux 2 PO(X, x) such that f(pcl(Ux)) Va(x). The family {Ux : x 2 K} is a cover of K by pre-open sets of X and hence there exists a finite subset K0 of K such that K [x2K 0 pclðU x Þ. Therefore, we obtain f ðKÞ [x2K 0 V aðxÞ . This shows that f(K) is N-closed relative to Y. h Corollary 6.3. Let f : X ! Y be an a.st.h.p.c. surjection. Then, the following properties hold: (a) If X is p-closed, then Y is nearly compact. (b) If X is countably p-closed, then Y is nearly countably compact.
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Theorem 6.4. If a function f : X ! Y has a pre-h-closed graph, then f(K) is h-closed in Y for each subset K which is pclosed relative to X. Proof. Let K be a p-closed relative to X and y 2 Y f(K). Then for each x 2 K we have (x,y) 62 G(f) and by Lemma 5.7 there exist Ux 2 PO(X, x) and an open set Vx of Y containing y such that f(pcl(Ux)) \ cl(Vx) = ;. The family {Ux : x 2 K} is a cover of K by pre-open sets of X. Since K is p-closed relative to X, there exists a finite subset K0 of K such that K [{pcl(Ux) : x 2 K0}. Put V = \{Vx : x 2 K0}. Then V is an open set containing y and f ðKÞ \ clðV Þ [x2K 0 f ðpclðU x ÞÞ \ clðV Þ [x2K 0 ½f ðpclðU x ÞÞ \ clðV x Þ ¼ ;. Therefore, we have y 2 h-cl(f(K)) and hence f(K) is h-closed in Y.
h
Theorem 6.5. Let X be a submaximal space. If a function f : X ! Y has a pre-h-closed graph, then f1(K) is h-closed in X for each subset K which is quasi-H-closed relative to Y. Proof. Let K be a quasi-H-closed set of Y and x 62 f1(K). Then for each y 2 K we have (x,y) 62 G(f) and by Lemma 5.7 there exists Uy 2 PO(X, x) and an open set Vy of Y containing y such that f(pcl(Uy)) \ cl(Vy) = ;. The family {Vy: y 2 K} is an open cover of K and there exists a finite subset K0 of K such that K [y2K 0 clðV y Þ. Since X is submaximal, each Uy is open in X and pcl(Uy) = cl(Uy). Set U ¼ \y2K 0 U y , then U is an open set containing x and f ðclðU ÞÞ \ clðKÞ [y2K 0 f ðclðU ÞÞ \ clðV y Þ [x2K 0 f ðpclðU y ÞÞ \ clðV y Þ ¼ ;. Therefore, we have cl(U) \ f1(K) = ; and hence x 62 h-cl(f1(K)). This shows that f1(K) is h-closed in X.
h
Corollary 6.6. Let X be a submaximal space and Y be a quasi-H-closed Hausdorff space. The following properties are equivalent for a function f : X ! Y: (a) f is a.st.h.p.c.; (b) G(f) is pre-h-closed in X · Y; (c) f is a.st.h-c. Proof (a) ) (b): This follows from Theorem 5.8. (b) ) (c): Let F be any regular closed set Y. By (2.2) of [28] F is quasi-H-closed and hence quasi-H-closed relative to Y. It follows from Theorem 6.5 that f1(F) is h-closed in X. Therefore, f is a.st.h-c. (c) ) (a): Obvious. h In conclusion, we study almost strongly h-precontinuity on topological spaces and obtain several improvements of results established by Noiri [23]. The generalizations of continuity related to some topological properties such as separation axioms, compactness, connectedness will been found to be useful in the study of topology. Thus we may stress the importance of almost strongly h-precontinuity as a branch of them and the possible application in digital topology, quantum physics, high energy physics and superstring theory.
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