International Journal of Mathematics and Mathematical Sciences Volume 2009, Article ID 465387,8pages
doi:10.1155/2009/465387
Research Article
On Semicompact Sets and Associated Properties
Mohammad S. Sarsak
Department of Mathematics, Faculty of Science, The Hashemite University, P.O. Box 150459, Zarqa 13115, Jordan
Correspondence should be addressed to Mohammad S. Sarsak,[email protected] Received 27 October 2009; Accepted 24 December 2009
Recommended by Naseer Shahzad
We continue the study of semicompact sets in a topological space. Several properties, mapping properties of semicompact sets are studied. A special interest toSCSspaces is given, where a spaceXisSCSif every subset ofXwhich is semicompact inXis semiclosed; we study several properties of such spaces, it is mainly shown that a semi-T2semicompact space isSCSif and only if it is extremally disconnected. It is also shown that in anos-regular spaceXif every point has an SCSneighborhood, thenXisSCS.
Copyrightq2009 Mohammad S. Sarsak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction and Preliminaries
A subsetAof a spaceX is called semi-open1ifA ⊂ IntA, or equivalently, if there exists an open subsetUofX such thatU ⊂A ⊂ U;Ais called semiclosed ifX \Ais semi-open.
The semiclosure sclAof a subsetAof a spaceXis 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 sclA A; it is also clear that ifA is a subset of a space X and x ∈ X, thenx ∈ sclAif and only ifS∩A /φ for each semi-open subset S of X containing x. A subset A of a spaceX is called preopen 2 resp., α-open 3if A⊂IntAresp.,A⊂Int IntA. Njastad3pointed out that the family of allα-open subsets of a spaceX, τ, denoted byτα, is a topology onXfiner thanτ. We will denote the families of semi-openresp., preopen,α-opensubsets of a spaceXbySOX resp.,POX,αOX. If X, τis a topological space, we will denote the spaceX, ταbyXα. Jankovi´c4pointed out thatPOX POXα,SOX SOXαandαOX αOXα. Reilly and Vamanamurthy observed in5thatτα SOX∩POX. It is known that the intersection of a semi-open resp., preopenset with anα-open set is semi-openresp., preopenand that the arbitrary union of semi-openresp., preopensets is semi-openresp., preopen.
A space X is called semicompact 6 resp., semi-Lindel ¨of 7 if any semi-open cover of X has a finite resp., countable subcover. A subsetAof a space X will be called semicompactresp., semi-Lindel ¨ofif it is semicompactresp., semi-Lindel ¨ofas a subspace.
A functionffrom a spaceXinto a spaceYis called semi-continuous1if the inverse image of each open subset ofY is semi-open inX, irresolute8if the inverse image of each semi-open subset ofYis semi-open inXandfis called pre-semi-openresp., pre-semiclosed 8if it maps semi-openresp., semiclosedsubsets ofXonto semi-openresp., semiclosed subsets ofY.
A spaceXis called semi-T29if for each distinct pointsxandyofX, there exist two disjoint semi-open subsetsUandVofXcontainingxandy,respectively.
A spaceX is called extremally disconnected10if the closure of each open subset of Xis open or equivalently if every regular closed subset ofXis preopen.
Throughout this paper, a spaceX stands for a topological space, and ifX is a space andA⊂X, thenAand IntAstand respectively for the closure ofAinX and the interior of AinX. For the concepts not defined here, we refer the reader to11.
In concluding this section, we recall the following facts for their importance in the material of our paper.
Proposition 1.1. LetA⊂B⊂X, whereXis a space. Then iIfAis semi-open inX, thenAis semi-open inB;
ii[12] IfAis semi-open inBandBis semi-open inX, thenAis semi-open inX.
Proposition 1.2. LetA⊂ B ⊂X, whereXis a space andBis preopen inX. ThenAis semi-open (resp., semiclosed) inBif and only ifAS∩B, whereSis semi-open (resp., semi-closed) inX.
2. Semicompact Sets
This section is mainly devoted to continue the study of semicompact sets. We also introduce and study semi-Lindel ¨of sets.
Definition 2.1see13. A subsetAof a spaceX is called semicompact relative toXif any semi-open cover ofAinXhas a finite subcover ofA.
By semicompact inX, we will mean semicompact relative toX.
Definition 2.2. A subsetAof a spaceXis called semi-Lindel ¨of inXif any semi-open cover of AinXhas a countable subcover ofA.
Remark 2.3. It is easy to see from the fact thatSOX SOXα, that a subsetAof a spaceXis semicompactresp., semi-Lindel ¨ofinXif and only if it is semicompactresp., semi-Lindel ¨of inXα.
The proof of the following proposition is straightforward, and thus omitted.
Proposition 2.4. The finite (resp., countable) union of semicompact (resp., semi-Lindel¨of) sets in a spaceXis semicompact (resp., semi-Lindel¨of) inX.
Proposition 2.5. LetB be a preopen subset of a spaceX andA ⊂ B. If Ais semicompact (resp., semi-Lindel¨of) inX, thenAis semicompact (resp., semi-Lindel¨of) inB.
Proof. We will show the case whenAis semicompact inX, the other case is similar. Suppose thatA{Aα:α∈Λ}is a cover ofAby semi-open sets inB. By Proposition1.2,AαSα∩B for eachα∈Λ, whereSαis semi-open inXfor eachα∈Λ. ThusS {Sα :α∈Λ}is a cover ofAby semi-open sets inX, butAis semicompact inX, so there existα1, α2, . . . , αn∈Λsuch thatA⊂in
i1Sαi, and thusA⊂in
i1Sαi∩B in
i1Aαi. Hence,Ais semicompact inB.
Corollary 2.6. LetAbe subset of a spaceX. IfAis semicompact (resp., semi-Lindel¨of) inX, thenA is semicompact (resp., semi-Lindel¨of).
Proposition 2.7. LetBbe a preopen subset of a spaceXandA⊂B. ThenAis semicompact (resp., semi-Lindel¨of) inXif and only ifAis semicompact (resp., semi-Lindel¨of) inB.
Proof. Necessity. It follows from Proposition2.5.
Sufficiency. We will show the case whenAis semicompact inB, the other case is similar.
Suppose thatS{Sα:α∈Λ}is a cover ofAby semi-open sets inX. ThenA{Sα∩B:α∈ Λ}is a cover ofA. SinceSαis semi-open inXfor eachα∈ΛandBis preopen inX, it follows from Proposition1.2thatSα∩Bis semi-open inBfor eachα∈Λ, butAis semicompact inB, so there existα1, α2, . . . αn∈Λsuch thatA⊂in
i1Sαi∩B⊂in
i1Sαi. Hence,Ais semicompact inX.
Corollary 2.8. A preopen subsetAof a spaceXis semicompact (resp., semi-Lindel¨of) if and only ifA is semicompact (resp., semi-Lindel¨of) inX.
Proposition 2.9. LetAbe a semicompact (resp., semi-Lindel¨of) set in a spaceX and Bbe a semi- closed subset ofX. ThenA∩Bis semicompact (resp., semi-Lindel¨of) inX. In particular, a semi-closed subsetAof a semicompact (resp., semi-Lindel¨of) spaceXis semicompact (resp., semi-Lindel¨of) inX.
Proof. We will show the case whenAis semicompact inX, the other case is similar. Suppose thatS {Sα : α ∈ Λ} is a cover of A∩B by semi-open sets inX. ThenA {Sα : α ∈ Λ} ∪ {X\B}is a cover ofAby semi-open sets inX, butAis semicompact inX, so there exist α1, α2, . . . , αn ∈Λsuch thatA⊂ in
i1Sαi∪X\B. ThusA∩B⊂ in
i1Sαi ∩B⊂ in
i1Sαi. Hence,A∩Bis strongly compact inX.
Proposition 2.10. Letf:X → Ybe an irresolute function. Then i[13] IfAis semicompact inX, thenfAis semicompact inY; iiIfAis semi-Lindel¨of inX, thenfAis semi-Lindel¨of inY.
Proof. iiThe proof is similar to that ofi. We will, however, show it for the convenience of the reader. Suppose thatS {Sα : α∈ Λ}is a cover offAby semi-open sets in Y. Then A{f−1Sα:α∈Λ}is a cover ofA, butfis irresolute, sof−1Sαis semi-open inXfor each α∈Λ. SinceAis semi-Lindel ¨of inX, there existα1, α2, α3, . . .∈Λsuch thatA⊂i∞
i1 f−1Sαi. ThusfA⊂i∞
i1 ff−1Sαi⊂i∞
i1 Sαi. Hence,fAis semi-Lindel ¨of inX.
Proposition 2.11. Let f : X → Y be a pre-semi-closed surjection. If for each y ∈ Y, f−1y is semicompact (resp., semi-Lindel¨of) inX, thenf−1Ais semicompact (resp., semi-Lindel¨of) inX wheneverAis semicompact (resp., semi-Lindel¨of) inY.
Proof. We will show the case whenAis semicompact inX, the other case is similar. Suppose that S {Sα : α ∈ Λ} is a cover of f−1A by semi-open sets in X. Then it follows
by assumption that for each y ∈ A, there exists a finite subcollection Sy of S such that f−1y ⊂
Sy. Let Vy ∪Sy. ThenVy is semi-open inX as any union of semi-open sets is semi-open. LetHy Y \fX\Vy. ThenHy is semi-open inY asf is pre-semi-closed, alsoy ∈ Hy for eachy ∈ Aasf−1y ⊂ Vy. Thus,H {Hy : y ∈ A} is a cover ofAby semi-open sets inY, butAis semicompact inY, so there existy1, y2, . . . , yn ∈ Asuch that A⊂in
i1Hyi. Thus,f−1A⊂in
i1f−1Hyi⊂in
i1Vyi. Since Syi is a finite subcollection ofS for eachi∈ {1,2, . . . , n}, it follows thatin
i1Syiis a finite subcollection ofS. Hence,f−1Ais semicompact inX.
3. SCS Spaces
Definition 3.1. A spaceX is said to beSCSif any subset ofXwhich is semicompact inX is semi-closed.
Remark 3.2. It follows from Remark2.3, that a spaceXisSCSif and only ifXαisSCS.
We recall the following result from3, it will be helpful to show the next two theorems.
Proposition 3.3. A spaceX is extremally disconnected if and only if the intersection of any two semi-open subsets ofXis semi-open.
Theorem 3.4. LetXbe a semi-T2extremally disconnected space. ThenXisSCS.
Proof. LetFbe a subset ofX which is semicompact inX and letx /∈F. Then for eachy ∈ F there exist two disjoint semi-open sets Uand V containing x and y respectively asX is semi-T2. SinceFis semicompact inX, there existy1, y2, . . . , yn∈Fsuch thatF ⊂n
i1Vyi. Let Un
i1Uyi. ThenUis a semi-open subset ofXthat containsxand disjoint fromFasXis extremally disconnected using Proposition3.3. Thus,x /∈sclF. 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. LetF A∪B,whereAandBare semi-closed inX. It follows by assumption thatA andBare semicompact in X and thus by Proposition2.4,F is semicompact inX, butX is SCS, soFis semi-closed inX. Hence by Proposition3.3,Xis extremally disconnected. The last part follows by Proposition2.9.
Corollary 3.6. For a semi-T2semicompact space, the followings are equivalent:
iXisSCS.
iiXis extremally disconnected.
Observing that a singleton of a spaceX is semi-open if and only if it is open, the following proposition seems clear.
Proposition 3.7. If every subset of a spaceXis semicompact inX, thenXisSCSif and only ifXis a finite discrete space.
Theorem 3.8. Letfbe a pre-semi-closed function from a spaceXonto a spaceY such that for each y∈Y,f−1yis semicompact inX. IfXisSCS, then so isY.
Proof. Let F be a semicompact set in Y. Then by Proposition 2.11, f−1F is semicompact inX, butX isSCS, so f−1Fis semi-closed inX, butf is a pre-semi-closed surjection, so Fff−1Fis semi-closed. Hence,Y isSCS.
Theorem 3.9. Let f be an irresolute one-to-one function from a spaceXinto anSCSspaceY. Then XisSCS.
Proof. LetF be a semicompact set inX. Then it follows from Proposition 2.10ithatfF is semicompact inY, but Y isSCS, so fF is semi-closed inY. Sincef is one-to-one and irresolute,F f−1fFis semi-closed inX. Hence,XisSCS.
Lemma 3.10. A subsetAof⊕Xαis semi-open if and only ifA∩Xα is semi-open inXαfor eachα.
Thus a subsetAof⊕Xαis semi-closed if and only ifA∩Xαis semi-closed inXαfor eachα.
Proof. SinceXαis open in⊕Xα, it follows that ifAis semi-open in⊕Xα, thenA∩Xαis semi- open in⊕Xαand thus semi-open inXαfor eachα. Now suppose thatA∩Xαis semi-open in Xαfor eachα. ThenA∩Xαis semi-open in⊕Xαfor eachαbecauseXαis open and thus semi- open in⊕Xα. Thus,A ∪A∩Xαis semi-open in⊕Xα as the arbitrary union of semi-open sets is semi-open.
Corollary 3.11. BeingSCSis hereditary with respect to preopen subsets.
Proof. LetAbe a preopen subset of anSCSspaceX and letB be semicompact inA. Then by Proposition 2.7, B is semicompact in X, but X is SCS, so B is semi-closed in X. By Proposition1.2,Bis semi-closed inA. Hence,AisSCS.
Corollary 3.12. ⊕XαisSCSif and only ifXαisSCSfor eachα.
Proof. Necessity. It follows from Corollary3.11sinceXαis open and thus preopen in⊕Xα. Sufficiency. Suppose thatXαis anSCSspace for eachαand letF be a subset of⊕Xα
which is semicompact in⊕Xα. SinceXαis closed and thus semi-closed in⊕Xα, it follows from Proposition2.9thatF∩Xα is semicompact in⊕Xα, butXαis preopen in⊕Xα, so it follows from Proposition2.7thatF∩Xαis semicompact inXα. SinceXαisSCS,F∩Xαis semi-closed inXαfor eachα, thus by Lemma3.10,Fis semi-closed in⊕Xα. Hence,⊕XαisSCS.
Recall that a spaceXis calleds-regular14if wheneverUis an open subset ofXand x ∈ U, there exists a semi-open subsetK of X and a semi-closed subsetS ofX such that x∈K ⊂S ⊂U. We now define a type of regularity which is stronger thans-regularity and weaker than regularity.
Definition 3.13. A spaceXis calledos-regular if wheneverUis an open subset ofXandx∈U, there exists an open subsetKofXand a semi-closed subsetSofXsuch thatx∈K⊂S⊂U.
Theorem 3.14. IfXis anos-regular space in which every point has anSCSneighborhood, thenXis SCS.
Proof. LetF be a subset ofX which is semicompact inX and letx /∈F. Then by assumption there exists anSCSneighborhood ofx. Since beingSCSis hereditary with respect to preopen
setsCorollary3.11, it follows thatxhas an openSCSneighborhood U. Now since X is os-regular, there exists an open subset K of X and a semi-closed subset S of X such that x∈K⊂S⊂U. SinceFis semicompact inXandSis a semi-closed subset ofX, it follows from Proposition2.9thatF∩Sis semicompact inX, thus by Proposition2.5,F∩Sis semicompact in U, butU isSCS, so F ∩S is semi-closed in U, that is, U\F ∩S is semi-open in U and thus semi-open inX by Proposition1.1iiasUis open and thus semi-open inX. Thus K∩U\F∩Sis a semi-open subset ofXthat containsxand disjoint fromFand therefore, x /∈sclF. Hence,Fis semi-closed inX, and therefore,XisSCS.
Corollary 3.15. IfX is a regular space in which every point has anSCSneighborhood, thenX is SCS.
Theorem 3.16. LetXbe a space in which every semi-closed subset is semicompact inX,Ybe anSCS space. Then any irresolute functionf fromX intoY is pre-semi-closed. In particular, any irresolute function from a semicompact spaceXinto anSCSspaceY is pre-semi-closed.
Proof. LetF be a semi-closed subset ofX. By assumption,Fis semicompact inX. Sincef is irresolute, it follows by Proposition2.10 thatfFis semicompact in Y. SinceY isSCS, it follows thatfFis semi-closed inY. The last part follows from Proposition2.9.
The following lemma will be helpful to show the next result, the easy proof is omitted.
Lemma 3.17. (i) The projection function is irresolute.
(ii) Letf : X → Y be irresolute and Abe anα-open subspace of X. Then the restriction functionf|A:A → Y is irresolute.
Theorem 3.18. LetXbe anSCSspace andYbe any space. Iff :X → Y is a function whose graph Gf is anα-open subspace ofX×Y in which every semi-closed subset is semicompact inGf, thenf is irresolute. In particular, any function having anSCSdomain and anα-open, semicompact graph is irresolute.
Proof. LetPX :X×Y → XandPY :X×Y → Y be the projection functions. SinceGf is an α-open subspace ofX×Y, it follows from Lemma3.17thatPX|Gfis irresolute. Thus it follows from Theorem3.16 thatPX|Gf is pre-semi-closed, that is,PX|Gf−1 is irresolute. Also,PY is irresolute. Thus,fPY◦PX|Gf−1is irresolute. The last part follows from Proposition2.9.
4. SLS Spaces
The study of this section is analogous to that of the preceding section, similar proofs are omitted.
Definition 4.1. A spaceX is said to beSLSif any subset ofXwhich is semi-Lindel ¨of inXis semi-closed.
Remark 4.2. It follows from Remark2.3, that a spaceXisSLSif and only ifXαisSLS.
Following Proposition 3.3, we will call a space X ω-extremally disconnected if the countable intersection of semi-open subsets ofXis semi-open.
Theorem 4.3. LetXbe a semi-T2 ω-extremally disconnected. ThenXisSLS.
Theorem 4.4. IfX is anSLSspace such that every semi-closed subsetAofX is semi-Lindel¨of in X, thenX isω-extremally disconnected. In particular, anSLSsemi-Lindel¨of space isω-extremally disconnected.
Corollary 4.5. For a semi-T2semi-Lindel¨of space, the followings are equivalent:
iXisSLS.
iiXisω-extremally disconnected.
Proposition 4.6. If every subset of a spaceXis semi-Lindel¨of inX, thenXisSLSif and only ifXis a countable discrete space.
Theorem 4.7. Letfbe a pre-semi-closed function from a spaceXonto a spaceY such that for each y∈Y,f−1yis semi-Lindel¨of inX. IfXisSLS, then so isY.
Theorem 4.8. Let f be an irresolute one-to-one function from a spaceX into anSLSspaceY. Then XisSLS.
Proposition 4.9. BeingSLSis hereditary with respect to preopen subsets.
Corollary 4.10. ⊕XαisSLSif and only ifXαisSLSfor eachα.
Theorem 4.11. IfXis anos-regular space in which every point has anSLSneighborhood, thenXis SLS.
Corollary 4.12. IfX is a regular space in which every point has anSLS neighborhood, thenX is SLS.
Theorem 4.13. LetXbe a space in which every semi-closed subset is semi-Lindel¨of inX,Ybe anSLS space. Then any irresolute functionf fromX intoY is pre-semi-closed. In particular, any irresolute function from a semi-Lindel¨of spaceXinto anSLSspaceY is pre-semi-closed.
Theorem 4.14. LetXbe anSLSspace andYbe any space. Iff:X → Y is a function whose graph Gf is anα-open subspace ofX×Y in which every semi-closed subset is semi-Lindel¨of inGf, thenfis irresolute. In particular, any function having anSLSdomain and anα-open, semi-Lindel¨of graph is irresolute.
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