Characterizations of Strongly Compact Spaces

Volume 2009, Article ID 573038,9pages doi:10.1155/2009/573038

Research Article

Characterizations of Strongly Compact Spaces

Ahmad Al-Omari,

_{Takashi Noiri,}

and Mohd. Salmi Md. Noorani

1_{Department of Mathematics and Statistics, Faculty of Science, Mu’tah University,}

P.O. Box 7, Karak 61710, Jordan

2_{2949-1 Shiokita-cho, Hinagu, Yatsushiro-shi, Kumamoto-ken 869-5142, Japan}

3_{School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,}

43600 Bangi, Selangor, Malaysia

Correspondence should be addressed to Ahmad Al-Omari,omarimutah1@yahoo.com

Received 4 July 2009; Accepted 30 September 2009

Recommended by Richard Wilson

A topological spaceX, τis said to be strongly compact if every preopen cover ofX, τadmits a finite subcover. In this paper, we introduce a new class of sets calledN-preopen sets which is weaker than both open sets andN-open sets. Where a subsetAis said to beN-preopen if for each

x∈Athere exists a preopen setUxcontainingxsuch thatUx−Ais a finite set. We investigate some properties of the sets. Moreover, we obtain new characterizations and preserving theorems of strongly compact spaces.

Copyrightq2009 Ahmad Al-Omari et al. 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

It is well known that the effects of the investigation of properties of closed bounded intervals

of real numbers, spaces of continuous functions, and solutions to differential equations

are the possible motivations for the formation of the notion of compactness. Compactness is now one of the most important, useful, and fundamental notions of not only general topology but also other advanced branches of mathematics. Many researchers have pithily studied the fundamental properties of compactness and now the results can be found in any undergraduate textbook on analysis and general topology. The productivity and fruitfulness of the notion of compactness motivated mathematicians to generalize this notion. In 1982,

Atia et al.1introduced a strong version of compactness defined in terms of preopen subsets

of a topological space which they called strongly compact. A topological space X is said

to be strongly compact if every preopen cover of X admits a finite subcover. Since then,

many mathematicians have obtained several results concerning its properties. The notion

2in 1984. They established several characterizations of these spaces. In 1987, Ganster3

obtained an interesting result that answered the question: what type of a space do we get when we take the one-point-compactification of a discrete space? He showed that this space is strongly compact. He proved that a topological space is strongly compact if and only if it

is compact and every infinite subset ofXhas nonempty interior. In 1988, Jankovi´c et al.4

showed that a topological spaceX, τis strongly compact if and only if it is compact and the

family of dense sets inX, τis finite. Quite recently Jafari and Noiri5,6, by introducing

the class of firmly precontinuous functions, found some new characterizations of strongly compact spaces. They also obtained properties of strongly compact spaces by using nets, filterbases, precomplete accumulation points. The notion of preopen sets plays an important role in the study of strongly compact spaces. In this paper, first we introduce and study the

notion of N-preopen sets as a generalization of preopen sets. Then, by using N-preopen

sets, we obtain new characterizations and further preservation theorems of strongly compact

spaces. We improve some of the results established by Mashhour et al.2. Throughout this

paper,X, τandY, σstand for topological spaces on which no separation axiom is assumed

unless otherwise stated. For a subsetAofX, the closure ofAand the interior ofAwill be

denoted by ClAand IntA, respectively. A subsetAof a topological spaceX is said to be

preopen7ifA ⊆ IntClA. A subsetAis said to beN-open8if for eachx ∈ Athere

exists an open setUx containing xsuch thatUx−Ais a finite set. The complement of an

N-open subset is said to beN-closed.

2.

N-Preopen Sets

In this section, we introduce and study the notion ofN-preopen sets.

Definition 2.1. A subsetAis said to beN-preopen if for eachx∈Athere exists a preopen set

Uxcontainingxsuch thatUx−Ais a finite set. The complement of anN-preopen subset is

said to beN-preclosed.

The family of allN-preopenresp., preopen, preclosedsubsets of a spaceX, τis

denoted byNPOX resp., POX, PCX.

Lemma 2.2. For a subset of a topological spaceX, τ, bothN-openness and preopenness implyN -preopenness.

For a subset of a topological space, the following implications hold:

open

preopen

N

-open

N

-preopen

2.1

Example 2.3. LetX 0,7be the closed interval with a topologyτ {φ, X,{1},{2},{1,2},

{1,2,3,5}. Then{2,3,5}is anN-open set which is not preopen.

Example 2.4. LetRbe the set of all real numbers with the usual topology. Then the setQof all

Lemma 2.5. Let X, τbe a topological space. Then the union of any family ofN-preopen sets is

N-preopen.

Proof. Let{Ui:i∈I}be a family ofN-preopen subsets ofXandx∈

i∈IUi. Thenx∈Ujfor

somej∈I. This implies that there exists a preopen subsetVofXcontainingxsuch thatV\Uj

is finite. SinceV \_i∈IUi⊆V\Uj, thenV \

i∈IUiis finite. Thus

i∈IUi ∈ NPOX.

Recall that a spaceX, τis called submaximal if every dense subset ofXis open.

Lemma 2.6see9. For a topological spaceX, τ, the followings are equivalent.

1Xis submaximal.

2Every preopen set is open.

Definition 2.7see10. A subsetAof a spaceXis said to beα-open ifA⊆IntClIntA.

Lemma 2.8see11. LetX, τbe a topological space. Then the intersection of anα-open set and a preopen set is preopen.

Theorem 2.9. Let X, τbe a submaximal topological space. Then X,NPOX is a topological space.

Proof. 1We haveφ, X∈ NPOX.

2 Let U, V ∈ NPOX and x ∈ U∩V. Then there exist preopen sets G, H ∈ X

containing xsuch that G\Uand H\V are finite. AndG∩H\U∩V G∩H∩

X\U∪X\V⊆G∩X\U∪H∩X\V. ThusG∩H\U∩Vis finite, and

sinceXis submaximal byLemma 2.6, the intersection of two preopen sets is preopen. Hence

U∩V ∈ NPOX.

3Let{Ui:i∈I}be any family ofN-preopen sets ofX. Then, byLemma 2.5,

i∈IUi

isN-preopen.

The converse of above theorem is not true.

Example 2.10. LetX{a, b, c}withτ {φ, X,{a},{a, b}}. ThenX,NPOXis a topological

space andX, τis not a submaximal topological space.

Lemma 2.11. LetX, τbe a topological space. Then the intersection of anα-open set and anN -preopen set isN-preopen.

Proof. LetUbeα-open andAN-preopen. Then for everyx ∈ A, there exists a preopen set

Vx⊆Xcontainingxsuch thatVx−Ais finite, and also byLemma 2.8,U∩Vxis preopen. Now

for eachx∈U∩A, there exists a preopen setU∩Vx⊆Xcontainingxand

U∩Vx−U∩A U∩Vx∩X−U∪X−A

U∩Vx∩X−U∪U∩Vx∩X−A

U∩Vx−A ⊆Vx−A.

2.2

The following lemma is well known and will be stated without the proof.

Lemma 2.12. A topological space is aT1-space if and only if every finite set is closed.

Proposition 2.13. If a topological spaceXis aT1-space, then every nonemptyN-preopen set contains

a nonempty preopen set.

Proof. LetAbe a nonempty N-preopen set andx ∈ A, then there exists a preopen setUx

containingxsuch thatUx−Ais finite. LetCUx−AUx∩X−A. Thenx∈Ux−C⊆A

and by Lemmas2.8and2.12,Ux−CUx∩X−Cis preopen.

The following example shows that ifXis not aT1-space, then there exists a nonempty

N-preopen set which does not contain a nonempty preopen set.

Example 2.14. LetX {a, b, c}withτ {φ, X,{a},{b},{a, b}}. Then{c}is anN-preopen set which does not contain any a nonempty preopen set.

Lemma 2.15see12. LetAandX0be subsets of a topological spaceX.

1IfA∈POXandX0∈SOX, thenA∩X0∈POX0.

2IfA∈POX0andX0∈POX, thenA∈POX.

Lemma 2.16. LetAandX0be subsets of a topological spaceX.

1IfA∈ NPOXandX0∈αOX, thenA∩X0∈ NPOX0.

2IfA∈ NPOX0andX0∈POX, thenA∈ NPOX.

Proof. 1 Let x ∈ A∩X0. Since Ais N-preopen in X, there exists a preopen set H of X

containingxsuch thatH−Ais finite. SinceX0isα-open, byLemma 2.8we haveH∩X0 ∈

POX. SinceX0 ∈ αOX ⊆ SOX, by Lemma 2.15,H∩X0 ∈ POX0,x ∈ H∩X0, and

H∩X0−A∩X0is finite. This shows thatA∩X0∈ NPOX0.

2IfA ∈ NPOX0, for eachx∈Athere existsVx ∈POX0containingxsuch that

Vx−Ais finite. SinceX0 ∈POX, byLemma 2.15,Vx ∈POXandVx−Ais finite and hence

A∈ NPOX.

Lemma 2.17. A subsetAof a spaceX isN-preopen if and only if for everyx ∈ A, there exist a preopen subsetUxcontainingxand a finite subsetCsuch thatUx−C⊆A.

Proof. LetAbeN-preopen andx ∈ A, then there exists a preopen subsetUx containingx

such thatUx−Ais finite. LetC Ux −A Ux∩X−A. ThenUx−C ⊆ A. Conversely,

letx∈A. Then there exist a preopen subsetUxcontainingxand a finite subsetCsuch that

Ux−C⊆A. ThusUx−A⊆CandUx−Ais a finite set.

Theorem 2.18. LetXbe a space andF ⊆X. IfFisN-preclosed, thenF⊆K∪Cfor some preclosed subsetKand a finite subsetC.

Proof. IfF isN-preclosed, thenX−F isN-preopen, and hence for everyx ∈ X−F, there

exists a preopen setUcontainingxand a finite set Csuch thatU−C ⊆ X−F. ThusF ⊆

X−U−C X−U∩X−C X−U∪C. LetKX−U. ThenKis a preclosed set such

3. Strongly Compact Spaces

Definition 3.1. 1A topological spaceXis said to be strongly compact1if every cover ofX

by preopen sets admits a finite subcover.

2A subset Aof a spaceX is said to be strongly compact relative toX 2if every

cover ofAby preopen sets ofXadmits a finite subcover.

Theorem 3.2. IfXis a space such that every preopen subset ofX is strongly compact relative toX, then every subset ofXis strongly compact relative toX.

Proof. LetBbe an arbitrary subset ofXand let{Ui :i∈I}be a cover ofBby preopen sets of

X. Then the family{Ui :i∈I}is a preopen cover of the preopen set∪{Ui :i∈I}. Hence by

hypothesis there is a finite subfamily{Uij :j ∈N0}which covers∪{Ui:i∈I}whereN0is a

finite subset of the naturalsN. This subfamily is also a cover of the setB.

Theorem 3.3. A subsetAof a spaceXis strongly compact relative toXif and only if for any cover

{Vα:α∈Λ}ofAbyN-preopen sets ofX, there exists a finite subsetΛ0ofΛsuch thatA⊆ ∪{Vα :

α∈Λ0}.

Proof. Let{Vα : α ∈ Λ}be a cover ofA and Vα ∈ NPOX. For eachx ∈ A, there exists

αx∈Λsuch thatx ∈Vαx. SinceVαx isN-preopen, there exists a preopen setUαxsuch

thatx∈UαxandUαx\Vαxis finite. The family{Uαx :x∈A}is a preopen cover ofA.

SinceAis strongly compact relative toX, there exists a finite subset, says,x1, x2, . . . , xnsuch

thatA⊆ ∪{Uαxi:i∈F}, whereF {1,2, . . . , n}. Now, we have

A⊆

i∈F

Uαxi\Vαxi∪Vαxi

i∈F

Uαxi\Vαxi

∪

i∈F

Vαxi

.

3.1

For eachxi,Uαxi\Vαxi is a finite set and there exists a finite subsetΛxiofΛsuch that

Uαxi\Vαxi∩A⊆ ∪{Vα:α∈Λxi}. Therefore, we haveA⊆i∈F∪{Vα :α∈Λxi}∪

_i∈FVαxi. HenceAis strongly compact relative toX.

Since every preopen set isN-preopen, the proof of the converse is obvious.

Corollary 3.4. For any spaceX, the following properties are equivalent:

1Xis strongly compact;

2everyN-preopen cover ofXadmits a finite subcover.

Theorem 3.5. For any spaceX, the following properties are equivalent:

1Xis strongly compact;

2every properN-preclosed set is strongly compact with respect toX.

Proof. 1⇒2LetAbe a properN-preclosed subset ofX. Let{Uα:α∈Λ}be a cover ofAby

preopen sets ofX. Now for eachx∈X−A, there is a preopen setVxsuch thatVx∩Ais finite.

Then{Uα:α∈Λ}∪{Vx:x∈X−A}is a preopen cover ofX. SinceXis strongly compact, there

thatX ∪{Uα :α∈Λ1}∪∪{Vxi : 1≤i≤n}; henceA⊂∪{Uα :α∈Λ1}∪∪{A∩Vxi :

1 ≤ i ≤ n}. SinceA∩Vxi is finite for eachi, there exists a finite subsetΛ2 ofΛ such that

∪{A∩Vxi : 1≤i≤n} ⊂ ∪{Uα :α∈Λ2}. Therefore, we obtainA⊂ ∪{Uα:α∈Λ1∪Λ2}. This

shows thatAis strongly compact relative toX.

2⇒1Let{Vα :α ∈Λ}be any preopen cover ofX. We choose and fix oneα0 ∈ Λ.

Then∪{Vα:α∈Λ− {α0}}is a preopen cover of aN-preclosed setX−Vα0. There exists a finite

subsetΛ0ofΛ− {α0}such thatX−Vα0⊂ ∪{Vα:α∈Λ0}. Therefore,X ∪{Vα:α∈Λ0∪ {α0}}.

This shows thatXis strongly compact.

Corollary 3.6see2. If a spaceX is strongly compact andAis preclosed, thenA is strongly compact relative toX.

Theorem 3.7. LetX, τbe a submaximal topological space. ThenX, τis strongly compact if and

only ifX,NPOXis compact.

Proof. Let {Vα : α ∈ Λ} be an open cover of X,NPOX. For each x ∈ X, there exists

αx∈Λsuch thatx∈Vαx. SinceVαxisN-preopen, there exists a preopen setUαxofX

such thatx ∈UαxandUαx\Vαxis finite. The family{Uαx :x ∈X}is a preopen cover

ofX, τ. SinceX, τis strongly compact, there exists a finite subset, says,x1, x2, . . . , xnsuch

thatX∪{Uαxi:i∈F}, whereF {1,2, . . . , n}. Now, we have

X

i∈F

Uαxi\Vαxi∪Vαxi

i∈F

Uαxi\Vαxi

∪

i∈F

Vαxi

.

3.2

For eachxi,Uαxi\Vαxi is a finite set and there exists a finite subsetΛxiofΛsuch that

Uαxi\Vαxi ⊆ ∪{Vα : α ∈ Λxi}. Therefore, we haveX

i∈F∪Vα : α ∈ Λxi∪

i∈FVαxi. HenceNPOXis compact.

Conversely, letUbe a preopen cover ofX, τ. ThenU ⊆ NPOX. SinceX,NPOX

is compact, there exists a finite subcover ofUforX. HenceX, τis strongly compact.

4. Preservation Theorems

Definition 4.1. A functionf :X → Y is said to beN-precontinuous iff−1_{VisN-preopen in}

Xfor each open setV inY.

Theorem 4.2. A function f : X → Y is N-precontinuous if and only if for each pointx ∈ X

and each open setV inY withfx ∈ V, there is anN-preopen setUinX such thatx ∈ Uand

fU⊆V.

Proof. Sufficiency. LetV be open inY andx∈f−1_{V. Thenfx∈V and thus there exists a}

Ux∈ NPOXsuch thatx∈UxandfUx⊆V. Thenx∈Ux ⊆f−1Vandf−1V ∪{Ux:

x∈f−1_{V}. Then byLemma 2.5,f}−1_{VisN-preopen.}

Necessity. Letx∈Xand letV be an open set ofY containingfx. Thenx∈f−1_V∈

Proposition 4.3. Iff:X → YisN-precontinuous andX0is anα-open set inX, then the restriction

f|X0 :X0 → Y isN-precontinuous.

Proof. SincefisN-precontinuous, for any open setV inY,f−1_{VisN-preopen inX. Hence}

byLemma 2.11, f−1_V∩X

0 isN-preopen in X. Therefore, by Lemma 2.16,f|X0

−1

V f−1_V∩X

0isN-preopen inX0. This implies thatf|X0isN-precontinuous.

Proposition 4.4. Letf:X → Ybe a function and let{Aα:α∈Λ}be anα-open cover ofX. If the

restrictionf|Aα :Aα:→ YisN-precontinuous for eachα∈Λ, thenfisN-precontinuous.

Proof. Suppose thatVis an arbitrary open set inY. Then for eachα∈Λ, we havef|Aα−1V

f−1V∩Aα ∈ NPOAαbecausef|Aα isN-precontinuous. Hence byLemma 2.16,f−1V∩

Aα ∈ NPOXfor eachα∈Λ. ByLemma 2.5, we obtain∪{f−1V∩Aα :α∈Λ}f−1V∈

NPOX. This implies thatfisN-precontinuous.

Theorem 4.5. Letfbe anN-precontinuous function from a spaceXonto a spaceY. IfXis strongly

compact, thenY is compact.

Proof. Let{Vα:α∈Λ}be an open cover ofY. Then{f−1Vα:α∈Λ}is anN-preopen cover

ofX. SinceX is strongly compact, byCorollary 3.4, there exists a finite subsetΛ0ofΛsuch

thatX∪{f−1_V

α:α∈Λ0}; henceY ∪{Vα:α∈Λ0}. ThereforeY is compact.

Definition 4.6see12. A functionf :X → Yis said to be precontinuous iff−1_{Vis preopen}

inXfor each open setV inY.

It is clear that every precontinuous function isN-precontinuous but not conversely.

Example 4.7. LetX {a, b, c, d},τ {φ, X,{a},{b},{a, b},{a, b, c}}, andσ {φ, X,{a, d},

{b, c}}. Let f : X, τ → X, σ be the identity function. Then f is an N

-precontinuous function which is not -precontinuous; because there exists{b, c} ∈σsuch that

f−1_{{b, c}/∈POX, τ.}

Corollary 4.8see2. Letf be a precontinuous function from a spaceX onto a spaceY. IfX is strongly compact, thenY is compact.

Definition 4.9see13. A functionf :X → Y is said to beM-preopen if the image of each

preopen setUofXis preopen inY.

Proposition 4.10. Iff : X → Y isM-preopen, then the image of anN-preopen set ofX isN -preopen inY.

Proof. Letf :X → Y beM-preopen andW anN-preopen subset ofX. For anyy∈ fW,

there existsx∈ W such thatfx y. SinceW isN-preopen, there exists a preopen setU

such thatx∈UandU−W Cis finite. Sincef isM-preopen,fUis preopen inY such

thaty fx ∈ fUandfU−fW ⊆ fU−W fCis finite. Therefore,fWis

N-preopen inY.

Definition 4.11see14. A functionf:X → Yis said to be preirresolute iff−1_{Vis preopen}

inXfor each preopen setVinY.

Proposition 4.12. Iff :X → Y is a preirresolute injection andAisN-preopen inY, thenf−1_A

Proof. Assume thatAis anN-preopen subset ofY. Letx∈f−1_{A. Thenfx∈Aand there}

exists a preopen setVcontainingfxsuch thatV−Ais finite. Sincefis preirresolute,f−1_V

is a preopen set containingx. Thusf−1_V−f−1_{A f}−1_{V −Aand it is finite. It follows}

thatf−1_{AisN-preopen inX.}

Definition 4.13. A functionf:X → Y is said to beN-preclosed iffAisN-preclosed inY for

each preclosed setAofX.

Theorem 4.14. Iff : X → Y is anN-preclosed surjection such thatf−1_{yis strongly compact}

relative toXfor eachy∈Y, andY is strongly compact, thenXis strongly compact.

Proof. Let{Uα:α∈Λ}be any preopen cover ofX. For eachy∈Y,f−1yis strongly compact

relative toXand there exists a finite subsetΛyofΛsuch thatf−1_{y⊂ ∪{U}

α :α∈Λy}.

Now we putUy ∪{Uα : α ∈ Λy}andVy Y −fX −Uy. Then, sincef isN

-preclosed,Vyis anN-preopen set inY containingy such thatf−1_{Vy ⊂ Uy. Since}

{Vy : y ∈ Y} is an N-preopen cover ofY, by Corollary 3.4 there exists a finite subset

{yk : 1≤ k ≤n} ⊆Y such thatY n_k1Vyk. Therefore,X f−1Y n_k1f−1Vyk ⊆

_n

k1Uyk

_n

k1{Uα:α∈Λyk}. This shows thatXis strongly compact.

Definition 4.15see2. A functionf :X → Y is said to beM-preclosed iffAis preclosed

inYfor each preclosed setAofX.

It is clear that everyM-preclosed function isN-preclosed but not conversely.

Example 4.16. LetX {a, b, c, d} with topologyτ {φ, X,{a},{b},{a, b},{a, b, c}}. Letf :

X, τ → X, τbe the function defined by settingfa c,fb d,fc a, andfd b.

Then f is an N-preclosed function which is not M-preclosed; because there exists {c} ∈

PCXsuch thatf{c} {a}/∈PCX.

Corollary 4.17see2. Iff :X → Y is anM-preclosed surjection such thatf−1_{yis strongly}

compact relative toXfor eachy∈Y, andYis strongly compact, thenXis strongly compact.

Definition 4.18. A functionf :X → Y is said to beδN-continuous if for eachx∈Xand each

preopen setV ofY containingfx, there exists anN-preopen setUofXcontainingxsuch

thatfU⊆V.

It is clear that every preirresolute function isδN-continuous but the converse is not

true.

Example 4.19. LetX {a, b, c, d}and τ {φ, X,{a},{b},{a, b},{a, b, c}}. Then the function

f :X, τ → X, τ, defined asfa c,fb d,fc a,andfd b, isδN-continuous

but it is not preirresolute.

Theorem 4.20. Letf : X → Y be aδN-continuous surjection from X ontoY. If X is strongly

compact, thenY is strongly compact.

Proof. Let{Vα:α∈Λ}be a preopen cover ofY. For eachx∈X, there existsαx∈Λsuch that

fx∈Vαx. SincefisδN-continuous, there exists anN-preopen setUαxofXcontainingx

such thatfUαx⊆Vαx. So{Uαx:x∈X}is anN-preopen cover of the strongly compact

spaceX, and by Corollary 3.4there exists a finite subset {xk : 1 ≤ k ≤ n} ⊆ X such that

X n_k1Uαxk. ThereforeY fX fn_k1Uαxk

_n

k1fUαxk ⊆

_n

k1Vαxk. This

Corollary 4.21see2. Letf : X → Y be a preirresolute surjection fromX ontoY. If X is strongly compact, thenY is strongly compact.

Acknowledgments

This work is financially supported by the Ministry of Higher Education, Malaysia under FRGS Grant no: UKM-ST-06-FRGS0008-2008. The authors also would like to thank the referees for useful comments and suggestions.

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