On Semicompact Sets and Associated Properties

We continue the study of semicompact sets in a topological space. Several properties, mapping properties of semicompact sets are studied. A special interest to SCS spaces is given, where a space X is SCS if every subset of X which is semicompact in X is semiclosed; we study several properties of such spaces, it is mainly shown that a semi-T2 semicompact space is SCS if and only if it is extremally disconnected. It is also shown that in an os-regular space X if every point has an SCS neighborhood, then X is SCS.


Introduction and Preliminaries
A subset A of a space X is called semi-open 1 if A ⊂ Int A, or equivalently, if there exists an open subset U of X such that U ⊂ A ⊂ U; A is called semiclosed if X \ A is semi-open.The semiclosure scl A of a subset A of a space X is the intersection of all semiclosed subsets of X that contain A or equivalently the smallest semiclosed subset of X that contains A.

Clearly, A is semiclosed if and only if scl A
A; it is also clear that if A is a subset of a space X and x ∈ X, then x ∈ scl A if and only if S ∩ A / φ for each semi-open subset S of X containing x.A subset A of a space X is called preopen 2 resp., α-open 3 if A ⊂ Int A resp., A ⊂ Int Int A .Njastad 3 pointed out that the family of all α-open subsets of a space X, τ , denoted by τ α , is a topology on X finer than τ.We will denote the families of semi-open resp., preopen, α-open subsets of a space X by SO X resp., PO X , αO X .If X, τ is a topological space, we will denote the space X, τ α by X α .Janković 4 pointed out that PO X PO X α , SO X SO X α and αO X αO X α .Reilly and Vamanamurthy observed in 5 that τ α SO X ∩ PO X .It is known that the intersection of a semi-open resp., preopen set with an α-open set is semi-open resp., preopen and that the arbitrary union of semi-open resp., preopen sets is semi-open resp., preopen .

International Journal of Mathematics and Mathematical Sciences
A space X is called semicompact 6 resp., semi-Lindel öf 7 if any semi-open cover of X has a finite resp., countable subcover.A subset A of a space X will be called semicompact resp., semi-Lindel öf if it is semicompact resp., semi-Lindel öf as a subspace.
A function f from a space X into a space Y is called semi-continuous A space X is called semi-T 2 9 if for each distinct points x and y of X, there exist two disjoint semi-open subsets U and V of X containing x and y, respectively.
A space X is called extremally disconnected 10 if the closure of each open subset of X is open or equivalently if every regular closed subset of X is preopen.
Throughout this paper, a space X stands for a topological space, and if X is a space and A ⊂ X, then A and Int A stand respectively for the closure of A in X and the interior of A in X.For the concepts not defined here, we refer the reader to 11 .
In concluding this section, we recall the following facts for their importance in the material of our paper.

Semicompact Sets
This section is mainly devoted to continue the study of semicompact sets.We also introduce and study semi-Lindel öf sets.Definition 2.1 see 13 .A subset A of a space X is called semicompact relative to X if any semi-open cover of A in X has a finite subcover of A.
By semicompact in X, we will mean semicompact relative to X.
The proof of the following proposition is straightforward, and thus omitted.

Proposition 2.5. Let B be a preopen subset of a space X and
Proof.We will show the case when A is semicompact in X, the other case is similar.Suppose Corollary 2.6.Let A be subset of a space X.If A is semicompact (resp., semi-Lindelöf) in X, then A is semicompact (resp., semi-Lindelöf).
Proposition 2.7.Let B be a preopen subset of a space X and Proof.Necessity.It follows from Proposition 2.5.
Sufficiency.We will show the case when A is semicompact in B, the other case is similar.
Corollary 2.8.A preopen subset A of a space X is semicompact (resp., semi-Lindelöf) if and only if A is semicompact (resp., semi-Lindelöf) in X. Proposition 2.9.Let A be a semicompact (resp., semi-Lindelöf) set in a space X and B be a semiclosed subset of X.Then A ∩ B is semicompact (resp., semi-Lindelöf) in X.In particular, a semi-closed subset A of a semicompact (resp., semi-Lindelöf) space X is semicompact (resp., semi-Lindelöf) in X.
Proof.We will show the case when A is semicompact in X, the other case is similar.Suppose that S {S α : Proof.ii The proof is similar to that of i .We will, however, show it for the convenience of the reader.Suppose that S {S α : Proof.We will show the case when A is semicompact in X, the other case is similar.Suppose that S {S α : α ∈ Λ} is a cover of f −1 A by semi-open sets in X.Then it follows

International Journal of Mathematics and Mathematical Sciences
by assumption that for each y ∈ A, there exists a finite subcollection S y of S such that Since S y i is a finite subcollection of S for each i ∈ {1, 2, . . ., n}, it follows that i n i 1 S y i is a finite subcollection of S. Hence, f −1 A is semicompact in X.

SCS Spaces
We recall the following result from 3 , it will be helpful to show the next two theorems.

Proposition 3.3. A space X is extremally disconnected if and only if the intersection of any two semi-open subsets of X is semi-open.
Theorem 3.4.Let X be a semi-T 2 extremally disconnected space.Then X is SCS.
Proof.Let F be a subset of X which is semicompact in X and let x / ∈ F. Then for each y ∈ F there exist two disjoint semi-open sets U and V containing x and y respectively as X is semi-T 2 .Since F is semicompact in X, there exist y 1 , y 2 , . . ., y n ∈ F such that F ⊂ n i 1 V y i .Let U n i 1 U y i .Then U is a semi-open subset of X that contains x and disjoint from F as X is extremally disconnected using Proposition 3.3 .Thus, x / ∈ scl F .Hence, F is semi-closed in X.
Theorem 3.5.If X is an SCS space such that every semi-closed subset A of X is semicompact in X, then X is extremally disconnected.In particular, an SCS semicompact space is extremally disconnected.
Proof.Let F A ∪ B, where A and B are semi-closed in X.It follows by assumption that A and B are semicompact in X and thus by Proposition 2.4, F is semicompact in X, but X is SCS, so F is semi-closed in X. Hence by Proposition 3.3, X is extremally disconnected.The last part follows by Proposition 2.9.Corollary 3.6.For a semi-T 2 semicompact space, the followings are equivalent: ii X is extremally disconnected.
Observing that a singleton of a space X is semi-open if and only if it is open, the following proposition seems clear.Proposition 3.7.If every subset of a space X is semicompact in X, then X is SCS if and only if X is a finite discrete space.Theorem 3.8.Let f be a pre-semi-closed function from a space X onto a space Y such that for each y Theorem 3.9.Let f be an irresolute one-to-one function from a space X into an SCS space Y .Then X is SCS.
Proof.Let F be a semicompact set in X.Then it follows from Proposition 2.10 i Sufficiency.Suppose that X α is an SCS space for each α and let F be a subset of ⊕X α which is semicompact in ⊕X α .Since X α is closed and thus semi-closed in ⊕X α , it follows from Proposition 2.9 that F ∩ X α is semicompact in ⊕X α , but X α is preopen in ⊕X α , so it follows from Proposition 2.7 that F ∩ X α is semicompact in X α .Since X α is SCS, F ∩ X α is semi-closed in X α for each α, thus by Lemma 3.10, F is semi-closed in ⊕X α .Hence, ⊕X α is SCS.
Recall that a space X is called s-regular 14 if whenever U is an open subset of X and x ∈ U, there exists a semi-open subset K of X and a semi-closed subset S of X such that x ∈ K ⊂ S ⊂ U. We now define a type of regularity which is stronger than s-regularity and weaker than regularity.Definition 3.13.A space X is called os-regular if whenever U is an open subset of X and x ∈ U, there exists an open subset K of X and a semi-closed subset S of X such that x ∈ K ⊂ S ⊂ U. Theorem 3.14.If X is an os-regular space in which every point has an SCS neighborhood, then X is SCS.
Proof.Let F be a subset of X which is semicompact in X and let x / ∈ F. Then by assumption there exists an SCS neighborhood of x.Since being SCS is hereditary with respect to preopen 1 if the inverse image of each open subset of Y is semi-open in X, irresolute 8 if the inverse image of each semi-open subset of Y is semi-open in X and f is called pre-semi-open resp., pre-semiclosed 8 if it maps semi-open resp., semiclosed subsets of X onto semi-open resp., semiclosed subsets of Y .
the arbitrary union of semi-open sets is semi-open.Being SCS is hereditary with respect to preopen subsets.Proof.Let A be a preopen subset of an SCS space X and let B be semicompact in A. Then by Proposition 2.7, B is semicompact in X, but X is SCS, so B is semi-closed in X.By Proposition 1.2, B is semi-closed in A. Hence, A is SCS.⊕X α is SCS if and only if X α is SCS for each α.Proof.Necessity.It follows from Corollary 3.11 since X α is open and thus preopen in ⊕X α .