ON p-CLOSED SPACES

We will continue the study of p-closed spaces. This class of spaces is strictly placed between the class of strongly compact spaces and the class of quasi-H-closed spaces. We will provide new characterizations of p-closed spaces and investigate their relationships with some other classes of topological spaces.


Introduction and preliminaries.
The aim of this paper is to continue the study of p-closed spaces, which were introduced by Abo-Khadra [1]. A topological space (X, τ) is called p-closed if every preopen cover of X has a finite subfamily whose pre-closures cover X.
Let A be a subset of a topological space (X, τ). Following Kronheimer [13], we call the interior of the closure of A, denoted by A + , the consolidation of A. Sets included in their consolidation play a significant role in, e.g., questions concerning covering properties, decompositions of continuity, etc. Such sets are called preopen [15] or locally dense [4]. A subset A of a space (X, τ) is called preclosed if its complement is preopen, i.e., if cl(int A) ⊆ A. The preclosure of A ⊆ X, denoted by pcl(A), is the intersection of all preclosed supersets of A. Since any union of preopen sets is also preopen, the preclosure of every set is preclosed. It is well known that pcl A = A ∪ cl(int A) for any A ⊆ X.
Another interesting property of preopen sets is the following: when a certain topological property is inherited by both open and dense sets, it is often then inherited by preopen sets.
Several important concepts in topology are and can be defined in terms of preopen sets. Among the most well known are Bourbaki's submaximal spaces (see [2]). A topological space is called submaximal if every (locally) dense subset is open or, equivalently, if every subset is locally closed, i.e., the intersection of an open set and a closed set. Another class of spaces commonly characterized in terms of preopen sets is the class of strongly irresolvable spaces introduced by Foran and Liebnitz in [9]. A topological space (X, τ) is called strongly irresolvable [9] if every open subspace of X is irresolvable, i.e., it cannot be represented as the disjoint union of two dense subsets. Subspaces that contain two disjoint dense subsets are called resolvable. Ganster [10] has pointed out that a space is strongly irresolvable if and only if every preopen set is semi-open, where a subset S of a space (X, τ) is called semi-open if S ⊆ cl(int S). We will denote the families of preopen (respectively, semi-open) sets of a space (X, τ) by PO(X) (respectively, SO(X)).
Many classical topological notions such as compactness and connectedness have been extended by using preopen sets instead of open sets. Among them are the class of strongly compact spaces [16] (= every preopen cover has a finite subcover) studied by Janković, Reilly and Vamanamurthy [12] and by Ganster [11], and the class of preconnected spaces (= spaces that cannot be represented as the disjoint union of two preopen subsets) introduced by Popa [19]. The study of topological properties via preopenness has gained significant importance in general topology and one example for that is the fact that four (out of the ten) articles in the 1998 Volume of "Memoirs of the Faculty of Science Kochi University Series A Mathematics" were more or less devoted to preopen sets.
A point x ∈ X is called a δ-cluster point of a set A [25] if A∩U = ∅ for every regular open set U containing x. The set of all δ-cluster points of A forms the δ-closure of A denoted by cl δ (A), and A is called δ-closed [25] if A = cl δ (A). If A ⊆ int(cl δ (A)), then A is said to be δ-preopen [21]. Complements of δ-preopen sets are called δ-preclosed and the δ-preclosure of a set A, denoted by δ-pcl(A), is the intersection of all δ-preclosed supersets of A.
Following [22], we will call a topological space (X, τ) δp-closed if for every δ-preopen Example 2.6. (i) Recall that a space (X, τ) is called α-scattered [7] if it has a dense set of isolated points. Clearly every α-scattered space is strongly irresolvable and so, by Theorem 2.2, every α-scattered QHC space is p-closed. In particular, the Katetov extension κN of the set of natural numbers N (cf. [20]) is p-closed and not compact, hence not strongly compact.
(ii) The unit interval [0, 1] with the usual topology is compact, hence QHC, but not p-closed since it is resolvable.
Then, X is p-closed and s-closed but not α-compact and hence not strongly compact (a space is α-compact if every cover by α-open sets has a finite subcover, where a set is α-open if it is the difference of an open and a nowhere dense set; clearly every α-open set is preopen but not vice versa). Additionally, this space is not δp-closed since every subset is δ-preopen.
We next discuss the relationship between p-closedness and compactness. Recall that a space (X, τ) is called nearly compact [24] if every cover of X by regular open sets has a finite subcover, i.e., the semiregularization (X, τ s ) of (X, τ) is compact. Example 4.8(d) in [20] shows that there exists a Hausdorff, non-compact, semi-regular and QHC space with a dense set of isolated points. Such a space is p-closed but not nearly compact. Example 2.10 in [22] provides another such example.
Proof. We first prove the case when the space is ℵ 0 -extremally disconnected. Let Hence, X is nearly compact. The proof of the second part of the theorem is similar to the first one and hence omitted.
On the other hand (cf. [20, page 450]) there exist dense-in-itself, compact and extremally disconnected Hausdorff spaces. Such spaces are resolvable and hence cannot be p-closed.
The preinterior of a set A, denoted by pint(A), is the union of all preopen subsets of A.
Theorem 2.8. For a topological space (X, τ) the following conditions are equivalent: (a) (X, τ) is p-closed, (b) every maximal filter base pre-θ-converges to some point of X, (c) every filter base pre-θ-accumulates at some point of X, x ∈ X} is a cover of X by preopen sets of X. By (a), there exists a finite number of points Therefore, we obtain F 0 = ∅. This is a contradiction.
(b)⇒(c). Let Ᏺ be any filter base on X. Then, there exists a maximal filter base Ᏺ 0 such that Ᏺ ⊆ Ᏺ 0 . By (b), Ᏺ 0 pre-θ-converges to some point x ∈ X. For every F ∈ Ᏺ and every This shows that Ᏺ pre-θ-accumulates at x.
This is a contradiction.
(d)⇒(a). Let {V α : α ∈ A} be a cover of X by preopen sets of X. Then {X \V α : α ∈ A} is a family of preclosed subsets of X such that ∩{X \V α : α ∈ A} = ∅. By (d), there exists a finite subset Definition 2.9. A topological space (X, τ) is said to be strongly p-regular (respectively, p-regular [8], almost p-regular [14]) if for each point x ∈ X and each preclosed set (respectively, closed set, regular closed set) F such that x ∈ F , there exist disjoint preopen sets U and V such that x ∈ U and F ⊆ V . Theorem 2.10. If a topological space X is p-closed and strongly p-regular (respectively, p-regular, almost p-regular), then X is strongly compact (respectively, compact, nearly compact).
Proof. We prove only the case of p-regular spaces. Let X be a p-closed and p- x ∈ X} is a preopen cover of the p-closed space X and hence there exists a finite amount of points, say, x 1 ,x 2 ,...,x n such that X = ∪ n i=1 pcl(U(x i )) = ∪ n i=1 V α(x i ) . This shows that X is compact.

p-closed subspaces.
Recall that a topological space (X, τ) is called hyperconnected if every open subset of X is dense. In the opposite case, X is called hyperdisconnected. A set A is called semi-regular [5] if it is both semi-open and semi-closed. Di Maio and Noiri [5] have shown that a set A is semi-regular if and only if there exists a regular open set U with U ⊆ A ⊆ cl(U ). Cameron [3] used the term regular semi-open for a semi-regular set.

Theorem 3.3. If every proper semi-regular subspace of a hyperdisconnected topological space (X, τ) is p-closed, then X is also p-closed.
Proof. Since (X, τ) is not hyperconnected, then there exists a proper semi-regular set A. Let {A i } i∈I be any preopen cover of X. Since A is semi-open, then by Lemma 3.1 A i ∩A = B i ∈ PO(A, τ | A). Then {B i } i∈I is a preopen cover of the p-closed space (A, τ | A). Then, there exists a finite subset F of I such that A = ∪ i∈F pcl A (B i ) ⊆ ∪ i∈F pcl(B i ) (by Lemma 3.2). Therefore, we have A ⊆ ∪ i∈F pcl(A i ). Since A is semi-regular, X \ A is also semi-regular and by a similar argument we can find a finite subset G of I such that X \ A ⊆ ∪ i∈G pcl(A i ). Hence, X = ∪ i∈F ∪G pcl(A i ). This shows that X is p-closed. A and X \ A are p-closed subspaces, then X is also p-closed. Proof. The proof is similar to the one of Theorem 3.3 and hence omitted.  If (X, τ) is a p-closed spaces and A is preregular (i.e., both preopen  and preclosed), then (A, τ | A) is also p-closed (as a subspace).

Theorem 3.4. If there exists a proper semi-regular subset A of a topological space (X, τ) such that
Proof. Let {A i } i∈I be any preopen cover of (A, τ | A). By Lemma 3.1, Since the only preopen set containing the excluded point is the whole space X, then the space in question is p-closed. However, the (infinite) set of isolated points of X is not p-closed.

Sets which are p-closed relative to a space.
A subset S of a topological space (X, τ) is said to be p-closed relative to X if for every cover {V α : α ∈ A} of S by preopen subsets of (X, τ), there exists a finite subset A 0 of A such that S ⊂ ∪{pcl(V α ) : α ∈ A 0 }. (a) S is p-closed relative to X, (b) every maximal filter base on X which meets S pre-θ-converges to some point of S, (c) every filter base on X which meets S pre-θ-accumulates at some point of S, (d) for every family {V α : α ∈ A} of preclosed subsets of (X, τ) such that [∩{V α : A point x ∈ X is said to be a pre-θ-accumulation point of a subset A of a topological space (X, τ) if pcl(U ) ∩ A = ∅ for every U ∈ PO(X, x). The set of all pre-θaccumulation points of A is called the pre-θ-closure of A and is denoted by pcl θ (A). A subset A of a topological space (X, τ) is said to be pre-θ-closed if pcl θ (A) = A. The complement of a pre-θ-closed set is called pre-θ-open. A be a subset A of a topological space (X, τ). Then:  Proof. Let {V α : α ∈ A} be any cover of X by pre-θ-open subsets of X. For each

Proposition 4.2. Let
. The family {V x : x ∈ X} is a preopen cover of X. Since X is p-closed, there exists a finite number of points, say, x 1 ,x 2 ,...,x n in X such that X = ∪{pcl(V x i ) : i = 1, 2,...,n}. Therefore, we obtain that X = ∪{V α(x i ) : i = 1, 2,...,n}. Theorem 4.5. Let A, B be subsets of a space X. If A is pre-θ-closed and B is p-closed relative to X, then A ∩ B is p-closed relative to X.
Proof. Let {V α : α ∈ A} be any cover of A∩B by preopen subsets of X.
is a cover of B by preopen sets of X. Since B is p-closed relative to X, there exist a finite number of points, say, x 1 ,x 2 ,...,x n in B \ A and a finite subset A 0 of A such that Corollary 4.6. If K is pre-θ-closed set of a p-closed space (X, τ), then K is p-closed relative to X. Question 4.7. If in a topological space (X, τ) every proper pre-θ-closed set is pclosed relative to X, is X necessarily p-closed?
A topological space (X, τ) is called preconnected [19] if X cannot be expressed as the union of two disjoint preopen sets. In the opposite case, X is called predisconnected. Note that every preconnected space is irresolvable but not vice versa.
Theorem 4.8. Let X be a predisconnected space. Then X is p-closed if and only if every preregular subset of X is p-closed relative to X.

Proof
Necessity. Every preregular set is pre-θ-closed by Proposition 4.2. Since X is pclosed, the proof is completed by Corollary 4.6.
Sufficiency. Let {V α : α ∈ A} be a preopen cover of X. Since X is predisconnected, there exists a proper preregular subset A of X. By our hypothesis, A and X \ A are pclosed relative to X. There exist finite subsets A 1 and A 2 of A such that Therefore, we obtain that Theorem 4.9. If there exists a proper preregular subset A of a topological space (X, τ) such that A and X \ A are p-closed relative to X, then X is p-closed.  (X, τ). If X 0 is a p-closed space, then it is p-closed relative to X.
Proof. Let {V α : α ∈ A} be any cover of X 0 by preopen subsets of X. Since X 0 ∈ SO(X), by Lemma 3.1, we have that X 0 ∩V α = W α ∈ PO(X 0 ) for each α ∈ A. Therefore, {W α : α ∈ A} is a preopen cover of X 0 . since X 0 is p-closed, there exists a finite subset A 0 of A such that X 0 = ∪{pcl X 0 (W α ) : α ∈ A 0 }. By Lemma 3.2, we obtain that X 0 ⊆ ∪{pcl(W α ) : α ∈ A 0 } ⊆ ∪{pcl(V α ) : α ∈ A 0 }. This shows that X 0 is p-closed relative to X. Theorem 4.11. Let X 0 be a preopen subset of a topological space (X, τ). If X 0 is a p-closed relative to X, then it is a p-closed subspace of X.
Proof. Let {V α : α ∈ A} be any cover of X 0 by preopen subsets of X 0 . Since X 0 ∈ PO(X), by Lemma 3.1, V α ∈ PO(X) for each α ∈ A. Since X 0 is p-closed relative to X, there exists a finite subset A 0 of A such that X 0 ⊆ ∪ {pcl(V α ) : α ∈ A 0 }. Since X 0 ∈ PO(X), by Lemma 3.5 we obtain X 0 = ∪{pcl X 0 (V α ) : α ∈ A 0 }. This shows that X 0 is a p-closed subspace of X. Proof. This is an immediate consequence of Theorems 4.10 and 4.11.
Proof. Let {V α : α ∈ A} be any cover of f (K) by preopen (respectively, open) subsets of Y . Since f is preirresolute (respectively, precontinuous), {f −1 (V α ) : α ∈ A} is a cover of K by preopen subsets of X, where K is p-closed relative to X. Therefore, there exists a finite subset A 0 of A such that K ⊆ ∪ α∈A 0 pcl(f −1 (V α )). Since f is preirresolute (respectively, precontinuous) and surjective, by Lemma 4.13, we have  (ii) If the product space α∈A X α is p-closed, then X α is p-closed for each α ∈ A. Remark 4.17. Even finite product of p-closed spaces need not be p-closed; for consider the product of the space from Example 2.6(i) with any two point indiscrete space. This product space shows that [1,Theorem 3.4.3] is wrong, i.e., every proper preregular subset might be p-closed relative to the space and still the space might fail to be p-closed. Additionally, [1, Example 3.4.1] is also false.