MATEMATIQKI VESNIK 48 (1996), 87–93

UDK 515.122.2

originalni nauqni rad research paper

ON P_{Σ} AND WEAKLY-P_{Σ} SPACES

M. Khan, T. Noiri and B. Ahmad

Abstract. In this paper, we point out that properties P_{Σ} due to Wang [18] and strongly s-regular due to Ganster [5] are equivalent to each other. We further study these spaces and weakly-P_{Σ} spaces defined by the second author [15].

1. Introduction

In 1981, Wang [18] defined a weak form of regularity called P_{Σ}. In 1984, the second author [13] defined the notion of weakly-P_{Σ} spaces which is weaker than that of P_{Σ} spaces. Recently, Ganster [5] has introduced the class of strongly sregular spaces which lies strictly between the class of regular spaces and the class of s-regular spaces in the sense of Maheshwari and Prasad [9]. In this paper, we point out that P_{Σ} and strongly s-regular are equivalent to each other. And we further investigate the properties of P_{Σ} and weakly-P_{Σ} spaces. Pre-almost open, pre-almost closed and regular-open functions are also defined and studied to obtain some preservation theorems of P_{Σ} and weakly-P_{Σ} spaces.

2. Preliminaries

Throughout this paper, spaces always mean topological spaces on which no separation axioms are assumed unless explicitly stated. Let X be a topological space and A be a subset of X. The closure of A and the interior of A in X are denoted by Cl(A) and Int(A), respectively. A subset A of X is said to be semiopen [8] if there exists an open subset U of X such that U ⊂ A ⊂ Cl(U). The complement of a semi-open set is said to be semi-closed. The semi-closure of A is defined as the intersection of all semi-closed sets containing A and is denoted by sCl(A). The semi-interior of A is defined as the union of all semi-open sets contained in A and is denoted by sInt(A). A subset A is said to be semi-regular [3] if it is semi-open and semi-closed. The family of all semi-open (resp. semi-regular) subsets of X is denoted by SO(X) (resp. SR(X)). A subset A is said to be preopen [10] if A ⊂ Int(Cl(A)). A subset A is said to be regular open (resp. regular closed) if A = Int(Cl(A)) (resp. A = Cl(Int(A))). The family of all regular open (resp. regular closed) subsets of X is denoted by RO(X) (resp. RC(X)). A point x of X is said to be in the θ-semiclosure [6] (resp. δ-closure [17]) of A, denoted by θ-sCl(A) (resp. Cl_{δ}(A)), if A ∩ Cl(U) ≠ ∅ (resp. A ∩ U ≠ ∅), for every U ∈ SO(X) (resp. U ∈ RO(X)) containing x. A subset A is said to be θ-semiclosed [6] (resp. δ-closed [17]) if θ-sCl(A) = A (resp. Cl_{δ}(A) = A).

AMS Subject Classification: 54D10, 54B05, 54C08

Keywords and phrases: P_{Σ}, weakly-P_{Σ}, strongly s-regular, s^{∗}-regular, s-regular, almostregular, regular closed, semi-open, almost-continuous, almost-open

Definition 1. A topological space X is said to be

(a) P_{Σ} [18] if every open subset of X is the union of regular closed sets;

(b) weakly-P_{Σ} [13] if every regular open subset of X is the union of regular closed sets;

(c) s^{∗}-regular [7] if for any semi-regular set A and any point x ∈ X − A, there exist disjoint open sets U and V such that A ⊂ U and x ∈ V;

(d) s-regular [9] (resp. semi-regular [4]) if for each closed (resp. semi-closed) set A and any point x ∈ X − A, there exist disjoint semi-open sets U and V such that A ⊂ U and x ∈ V;

(e) extremally disconnected (briefly E.D.) if Cl(U) is open in X, for every open set U in X;

(f) almost regular [15] if for any regular closed set A and any point x ∈ X − A, there exist disjoint open sets U and V such that A ⊂ U and x ∈ V;

(g) strongly s-regular [5] if for each closed set A and any point x ∈ X − A, there exists an F ∈ RC(X) such that x ∈ F and F ∩ A = ∅.

3. P_{Σ} and weakly P_{Σ} spaces

First of all, we point out that P_{Σ} and strongly s-regular are equivalent to each other.

Theorem 1. (Ganster [5]) The following are equivalent for a topological space X:

(a) X is P_{Σ}.

(b) For any open subset U of X and any point x ∈ U there exists an F ∈ RC(X) such that x ∈ F ⊂ U.

(c) X is strongly s-regular.

Theorem 2. The following are equivalent for a topological space X:

(a) X is weakly-P_{Σ}.

(b) For any regular open subset U of X and any point x ∈ U, there exists an F ∈ RC(X) such that x ∈ F ⊂ U.

(c) Every regular closed set in X is the intersection of regular open sets.

(d) θ-sCl(A) ⊂ Cl_{δ}(A) for every subset A of X.

(e) Every δ-closed set of X is θ-semiclosed in X.

On P_{Σ} and weakly P_{Σ} spaces 89

Proof. The proof is quite similar to that of [5, Theorem 1] and is thus omitted.

In [5, Theorem 2], Ganster showed that strong s-regularity is open hereditary. We shall improve this result in the following theorem.

Theorem 3. If X is a P_{Σ} spaces and Y is preopen in X, then the subspace Y is P_{Σ}.

Proof. Let U be an arbitrary open susbet of Y. Then there exists an open subset V of X such that U = V ∩ Y. Since X is a P_{Σ} space, we have V = ⋃{ V_{α} : α ∈ ∇ }, where V_{α} ∈ RC(X) for each α ∈ ∇. It is easily checked that a subset is regular closed in X if and only if it is closed and semi-open in X. Therefore, V_{α} ∩Y is closed in Y for each α ∈ ∇. By [14, Lemma 2.2], we obtain V_{α} ∩ Y ∈ SO(Y) and hence V_{α} ∩ Y ∈ RC(Y) for each α ∈ ∇ and U = ⋃{ V_{α} ∩ Y : α ∈ ∇ }. This shows that the subspace Y is P_{Σ}.

Corollary 1. (Ganster [5]) Strong s-regularity is open hereditary.

Theorem 4. If X is a weakly-P_{Σ} space and Y is open in X, then the subspace Y is weakly-P_{Σ}.

Proof. Let U be an arbitrary regular open subset of Y. It is shown in [11, Lemma 3] that Int_{Y} (Cl_{Y} (A)) = Y ∩ Int(Cl(A)) for any open subset Y of X and any subset A of Y. Therefore, there exists a regular open subset V of X such that U = Y ∩ V. Since X is a weakly-P_{Σ} space, we have V = ⋃{ V_{α} : α ∈ ∇ }, where V_{α} ∈ RC(X) for each α ∈ ∇. Similarly to the proof of Theorem 3, we obtain V_{α} ∩Y is regular closed in Y for each α ∈ ∇ and U = ⋃{ V_{α} ∩ Y : α ∈ ∇ }. This shows that the subspace Y is weakly-P_{Σ}.

Theorem 5. If a space X is s-regular and s^{∗}-regular, then it is regular.

Proof. Let U be any open subset of X and x ∈ U. Since X is s-regular, there exists G ∈ SO(X) such that x ∈ G ⊂ sCl(G) ⊂ U [9, Theorem 2]. It follows from Proposition 2.2 of [3] that sCl(G) ∈ SR(X). Since X is s^{∗}-regular, there exists an open subset O of X such that x ∈ O ⊂ Cl(O) ⊂ sCl(G) [7, Theorem 1]; hence x ∈ O ⊂ Cl(O) ⊂ U. This shows that X is regular.

Since RC(X) ⊂ SR(X), every s^{∗}-regular space is almost regular. Ganster showed that there exists a Hausdorff strongly s-regular space which is not almost regular [5, Example 4]. By these results and Example 1 (stated below), we obtain the following property.

Remark 1. s^{∗}-regularity is independent of strong s-regularity and also sregularity.

Example 1. Let X = {a, b, c} and τ = {∅, {a}, {c}, {a, c}, {b, c}, X}. Then (X, τ) is an s^{∗}-regular space. And it is not s-regular since a subset {a, b} is closed and not semi-open in (X, τ).

Theorem 6. A topological space X is s^{∗}-regular if and only if it is E.D.

Proof. Necessity. Let X be s^{∗}-regular and V a nonempty open set in X. Then we have Cl(V) ∈ RC(X) and RC(X) ⊂ SR(X). For each x ∈ Cl(V), there exists an open set U_{x}, such that x ∈ U_{x} ⊂ Cl(U_{x}) ⊂ Cl(V). Therefore, Cl(V) = ⋃{ U_{x} : x ∈ Cl(V) } is open in X. This shows that X is E.D.

Sufficiency. Let X be E.D. and A be any semi-regular set in X. Since A is semi-open, by [3, Proposition 2.4] we have sCl(A) = Cl(A) and hence A = sCl(A) = Cl(A) = Cl(Int(A)). This shows that A is open and closed in X. Therefore, X is s^{∗}-regular.

Corollary 2. (Ganster [5]) The following are equivalent for an E.D. space X:

(a) X is regular.

(b) X is strongly s-regular.

(c) X is s-regular.

Proof. The proof follows immediately from Theorems 5 and 6.

We have the following diagram related to separation axioms defined in §2.

extremally disconnected  regular  semi-regular
↑↓  ↓  ↓
s^{∗}-regular  strongly s-regular → s-regular
↓  ↓
almost regular  →  weakly-P_{Σ}

Remark 2. None of implications in the above diagram is reversible as shown by the following:

(a) Dorsett [4] pointed out that semi-regularity is independent of regularity and is strictly stronger than s-regularity.

(b) In Examples 1 and 2 of [5], Ganster showed that strong s-regularity lies strictly between regularity and s-regularity. We should note that the term “semiregular” in [5, Example 2] is different from “semi-regular” in the sense of Dorsett [4].

(c) By [5, Example 4] and Example 1, the both notions of almost regular and strongly s-regular are strictly stronger than that of weakly-P_{Σ}.

(d) The real numbers with the usual topology is a regular space which is not E.D. Therefore, by Example 1 “E.D.” and “regular” are independent of each other. And also almost regularity does not always imply s^{∗}-regularity.

In [5, Theorem 3], Ganster showed that strong s-regularity is productive. We obtain the similar result about weakly-P_{Σ} spaces.

Theorem 7. If (X_{α}, τ_{α}) is a weakly-P_{Σ} space for each α ∈ ∇, then the product space (X, τ) = ∏{ (X_{α}, τ_{α}) : α ∈ ∇ } is weakly-P_{Σ}.

Proof. Let W be an arbitrary regular open set in (X, τ) and x ∈ W. Then we have x ∈ ∏{ U_{α} : α ∈ ∇ } ⊂ W, where U_{α} is open in (X_{α}, τ_{α}) for each α ∈ ∇ and there exists a finite subset ∇_{0} of ∇ such that U_{α} = X_{α} whenever α ∈ ∇ − ∇_{0}.

Therefore, we have x ∈ ∏{ Int(Cl(U_{α})) : α ∈ ∇ } ⊂ W. For each α ∈ ∇_{0}, we have x_{α} ∈ Int(Cl(U_{α})) and hence x_{α} ∈ F_{α} ⊂ Int(Cl(U_{α})) for some F_{α} ∈ RC(X_{α}, τ_{α}). Now, let F = ∏{ F_{α} : α ∈ ∇_{0} } × ∏{ X_{α} : α ∈ ∇ − ∇_{0} }. Then, we obtain F ∈ RC(X, τ) and x ∈ F ⊂ ∏{ Int(Cl(U_{α})) : α ∈ ∇ } ⊂ W. This shows that (X, τ) is weakly-P_{Σ}.

4. Preservation theorems

We shall recall definitions of some functions used in the sequel to obtain several preservation theorems.

Definition 2. A function f: X → Y is said to be:

(a) almost-continuous [16] if f^{1}(V) is open in X for every V ∈ RO(Y);

(b) completely-continuous [1] (resp. R-map [2]) if f^{1}(V) ∈ RO(X) for every open subset V of Y (resp. V ∈ RO(Y));

(c) almost-open [16] if f(U) is open in Y for every U ∈ RO(X).

Definition 3. A function f: X → Y is said to be:

(a) pre-almost open (resp. regular open) if f(U) ∈ RO(Y) for every U ∈ RO(X) (resp. open set U in X);

(b) pre-almost closed if f(U) ∈ RC(Y) for every U ∈ RC(X).

Lemma 1. (Noiri [12]) Every almost-continuous almost-open function is an R-map.

Theorem 8. If f: X → Y is an almost-continuous and open (resp. almostopen) injection and Y is P_{Σ}, then X is P_{Σ} (resp. weakly-P_{Σ}).

Proof. Let U be an arbitrary open (resp. regular open) subset of X. Then f(U) is open in Y and f(U) = ⋃{ V_{α} : α ∈ ∇ }, where V_{α} ∈ RC(Y) for each α ∈ ∇. Since f is injective, we have U = ⋃{ f^{1}(V_{α}) : α ∈ ∇ }. By Lemma 1, f is an R-map and hence f^{1}(V_{α}) ∈ RC(X) for each α ∈ ∇. Therefore, X is P_{Σ} (resp. weakly-P_{Σ}).

Theorem 9. If f: X → Y is an almost-continuous and pre-almost open (resp. regular open) injection and Y is weakly-P_{Σ}, then X is weakly-P_{Σ} (resp. P_{Σ}).

Proof. Let U be an arbitrary regular open (resp. open) set in X. Then f(U) is regular open in Y and f(U) = ⋃{ V_{α} : α ∈ ∇ }, where V_{α} ∈ RC(Y) for each α ∈ ∇. Since f is injective, we have U = ⋃{ f^{1}(V_{α}) : α ∈ ∇ }. Every regular open function is pre-almost open and every pre-almost open function is almost open. Therefore, by Lemma 1, f is an R-map and hence f^{1}(V_{α}) ∈ RC(X) for each α ∈ ∇. This shows that X is weakly-P_{Σ} (resp. P_{Σ}).

Theorem 10. If f: X → Y is a continuous (resp. completely continuous) and pre-almost closed surjection and X is P_{Σ} (resp. weakly-P_{Σ}), then Y is P_{Σ}.

Proof. Let V be an arbitrary open subset of Y. Then f^{1}(V) is open (resp. regular open) in X. Since X is P_{Σ} (resp. weakly-P_{Σ}), f^{1}(V) = ⋃{ U_{α} : α ∈ ∇ }, where U_{α} ∈ RC(X) for each α ∈ ∇. Since f is surjective, we have V = ⋃{ f(U_{α}) : α ∈ ∇ }. Since f is pre-almost closed, f(U_{α}) ∈ RC(Y) for each α ∈ ∇. This shows that Y is P_{Σ}.

Theorem 11. If f: X → Y is an R-map (resp. almost-continuous) and prealmost closed surjection and X is weakly-P_{Σ} (resp. P_{Σ}), then Y is weakly-P_{Σ}.

Proof. Let V be an arbitrary regular open set in Y. Then f^{1}(V) is regular open (resp. open) in X. Since X is weakly-P_{Σ} (resp. P_{Σ}), f^{1}(V) = ⋃{ U_{α} : α ∈ ∇ }, where U_{α} ∈ RC(X) for each α ∈ ∇. Since f is a pre-almost closed surjection, we have V = ⋃{ f(U_{α}) : α ∈ ∇ } and f(U_{α}) ∈ RC(Y) for each α ∈ ∇. This shows that Y is weakly-P_{Σ}.

Lemma 2. If f: X → Y is a pre-almost open function, then for any point y of Y and any A ∈ RC(X) containing f^{1}(y), there exists a B ∈ RC(Y) containing y such that f^{1}(B) ⊂ A.

Proof. Let B = Y − f(X − A). Then, since f^{1}(y) ⊂ A, it follows that y ∈ B and B ∈ RC(Y) because f is pre-almost open. By a straightforward calculation, we have f^{1}(B) ⊂ A.

A subset S of a space X is said to be S-closed relative to X [13] if for every cover { U_{α} : α ∈ ∇ } of S by semi-open sets of X there exists a finite subset ∇_{0} of ∇ such that S ⊂ ⋃{ Cl(U_{α}) : α ∈ ∇_{0} }. It is obvious that a subset S of a space X is S-closed relative to X if and only if every cover of S by regular closed sets in X has a finite subcover.

Theorem 12. Let f: X → Y be a continuous and pre-almost open surjection such that f^{1}(y) is S-closed relative to X for each point y of Y. If X is P_{Σ} (resp. weakly-P_{Σ}), then Y is P_{Σ} (resp. weakly-P_{Σ}).

Proof. Let V be an arbitrary open (resp. regular open) set in Y and y ∈ V. Then, since pre-almost open sets are almost open, by Lemma 1 f^{1}(V) is open (resp. regular open) in X. Since X is P_{Σ} (resp. weakly-P_{Σ}) and f^{1}(y) ⊂ f^{1}(V), for each x ∈ f^{1}(y), there exists R(x) ∈ RC(X) such that x ∈ R(x) ⊂ f^{1}(V). Since the family { R(X) : x ∈ f^{1}(y) } is a regular closed cover of f^{1}(y) and f^{1}(y) is S-closed relative to X, there exists a finite number of points, say x_{1}, x_{2}, …, x_{n}, such that f^{1}(y) ⊂ ⋃{ R(x_{i}) : 1 ⩽ i ⩽ n }. The finite union of regular closed sets is regular closed. Therefore, by Lemma 2 there exists R ∈ RC(Y) containing y such that f^{1}(R) ⊂ ⋃{ R(x_{i}) : 1 ⩽ i ⩽ n }, where each R(x_{i}) is contained in f^{1}(V). Therefore, we obtain that y ∈ R ⊂ V and Y is P_{Σ} (resp. weakly-P_{Σ}).

REFERENCES

[1] S. P. Arya and R. Gupta, On strongly continuous mappings, Kyungpook Math. J. 14 (1974), 131–143

[2] D. A. Carnahan, Some Properties Related to Compactness in Topological Spaces, Ph. D. Thesis, Univ. Arkansas, 1973 [3] G. Di Maio and T. Noiri, On s-closed spaces, Indian J. Pure Appl. Math. 18 (1987), 226–233 [4] C. Dorsett, Semi-regular spaces, Soochow J. Math. 8 (1982), 45–53

[5] M. Ganster, On strongly s-regular spaces, Glasnik Mat. 25(45) (1990), 195–201

[6] J. E. Joseph and M. H. Kwack, On S-closed spaces, Proc. Amer. Math. Soc. 80 (1980), 341–348

[7] M. Khan, T. Noiri and B. Ahmad, On s^{∗}-regular and extremally disconnected spaces (submitted)

[8] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36–41

[9] S. N. Maheshwari and R. Prasad, On s-regular spaces, Glasnik Mat. 10(30) (1975), 347–350 [10] A. S. Mashour, M. E. Abd El-Monsef and S. N. El-Deeb, On precontinuous and weak pre-

continuous mappings, Proc. Math. Phys. Soc. Egypt 53 (1982), 47–53

[11] T. Noiri, On almost-open mappings, Mem. Miyakonojo Tech. Coll. 7 (1972), 167–171

[12] T. Noiri, Almost-continuity and some separation axioms, Glasnik Mat. 9(29) (1974), 131– 135

[13] T. Noiri, A note on S-closed spaces, Bull. Inst. Math. Acad. Sinica 12 (1984), 229–235

[14] T. Noiri and B. Ahmad, A note on semi-open functions, Math. Sem. Notes Kobe Univ. 10 (1982), 437–441

[15] M. K. Singal and S. P. Arya, On almost-regular spaces, Glasnik Mat. 4(24) (1969), 89–99 [16] M. K. Singal and A. R. Singal, Almost-continuous mappings, Yokohama Math. J. 16 (1968),

63–73

[17] N. V. Veličko, H-closed topological spaces, Amer. Math. Soc. Transl. (2) 78 (1968), 103–118 [18] G. J. Wang, On S-closed spaces, Acta Math. Sinica 24 (1981), 55-63

(received 26.04.1996.)

Department of Mathematics, Goverment College, Rajanpur, Pakistan Department of Mathematics, Yatsushiro College of Technology, Yatsushiro, Kumamoto, 866 Japan Department of Mathematics, Centre of Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan