Fuzzy Inverse Compactness

Volume 2008, Article ID 436570,9pages doi:10.1155/2008/436570

Research Article

Fuzzy Inverse Compactness

Halis Ayg ¨un,1 A. Arzu Bural,2and S. R. T. Kudri3

1_{Department of Mathematics, Faculty of Arts and Sciences, Kocaeli University, Kocaeli 41380, Turkey} 2_{Department of Mathematics, Faculty of Education, Kocaeli University, Kocaeli 41380, Turkey} 3_{Department of Mathematics, Federal University of Parana, P.O. Box 019081, Curitiba,}

PR 81531-990, Brazil

Correspondence should be addressed to Halis Ayg ¨un,[email protected]

Received 9 February 2007; Revised 25 November 2007; Accepted 14 January 2008

Recommended by Heinz Gumm

We introduce definitions of fuzzy inverse compactness, fuzzy inverse countable compactness, and fuzzy inverse Lindel ¨ofness on arbitraryL-fuzzy sets inL-fuzzy topological spaces. We prove that the proposed definitions are good extensions of the corresponding concepts in ordinary topology and obtain different characterizations of fuzzy inverse compactness.

Copyrightq2008 Halis Ayg ¨un 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

In ordinary topology, Matveev1has introduced a topological property called inverse com-pactness which is weaker than comcom-pactness and stronger than countable comcom-pactness. In1,2, inverse countable compactness and inverse Lindel ¨ofness have been defined and studied.

A topological spaceX is called inversely compact if and only if for every open cover

β ofX, one can select a finite subcoverγ ofX which consists of the elements ofβ or their complementsbut of courseγ is prohibited to contain bothUandX\Ufor anyU∈β.

InL-fuzzy topological spaces fuzzy compactness has been introduced by Warner and McLean3and extended to arbitraryL-fuzzy sets by Kudri4.

In this paper, we initiate fuzzy inverse compactness inL-fuzzy topological spaces which is weaker than fuzzy compactness and introduce fuzzy inverse countable compactness and fuzzy inverse Lindel ¨ofness. We prove that proposed definitions are good extensions of the corresponding notions in ordinary topology.

2. Preliminaries

Throughout this paperXandY will be nonempty ordinary sets andLL≤,∨,∧,will denote a completely distributive lattice with a smallest element 0 and a largest element 10/1 and with an order-reversing involutiona→aa ∈L.τ andTwill denoteL-fuzzy topology and ordinary topology, respectively.

Definition 2.1see 5. An element p of a latticeLis called prime if and only if p / 1 and whenevera, b∈Lwitha∧b≤pthena≤porb≤p.

The set of all prime elements of a latticeLwill be denoted by prL.

Definition 2.2see5. An elementαof a latticeLis called coprimeor union irreducibleif and only ifα /0 and whenevera, b∈Lwithα≤a∨bthenα≤porα≤b.

The set of all coprime elements of a latticeLwill be denoted byML.

From the definitions we have thatp∈prL⇔p∈ML.

Definition 2.3see3. LetLbe a complete lattice. A subsetUofLis called Scott open if and only if it is an upper set and is inaccessible by directed joins, that is,

aifa∈Uanda≤b, thenb∈U;

bifDis a directed subset ofLwithD∈U, then there is ad∈Dwithd∈U.

The collection of all Scott-open subsets ofLis a topology onLand is called Scott topology of

L. The Scott topology of completely distributive latticeLis generated by the sets of the form {x∈L |x p}, wherep ∈prL. This means that the family{{x ∈L |x p} | p ∈prL}

forms a base for Scott topology onL.

A functionf fromX, TtoLwith its Scott topology is Scott continuous if and only if

f−1_{{x∈X|xp}∈Tfor everyp∈prL.}

Definition 2.4see6. AnL-fuzzy topology onXis a subsetτ ofLX_{satisfying the following}

properties:

ithe constant functions 0 and 1 belong toτ;

iiiff, g∈τ, thenf∧g∈τ;

iiiif{fi:i∈J} ⊆τ, theni∈Jfi∈τ.

The pairX, τis called anL-fuzzy topological space. The elements ofτare called openL-fuzzy sets. AnL-fuzzy sethis said to be closed if and only ifh∈τ.

Definition 2.5 see 6. LetX, τ and Y, τ∗ be L-fuzzy topological spaces. A function ϕ :

X, τ→Y, τ∗is called fuzzy continuous if and only ifϕ−1_{g∈τfor everyg∈τ}∗_.ϕ−1_g gϕx.

Definition 2.6see7. LetX, Tbe an ordinary topological space. The set of all Scott continu-ous functions fromX, TtoLwith its Scott topology forms anL-fuzzy topology onX, which will be denoted byωLT, that is,

ωLT

Definition 2.7 see 7. Let X, T be an ordinary topological space. An L-fuzzy topologi-cal property Pf is a ”good extension” of a topological property P if and only if the

topo-logical space X, T has P if and only if the inducedL-fuzzy topological space X, ωLT

hasPf.

Definition 2.8see4. LetX, τbe anL-fuzzy topological space,g ∈LX_{andp∈L. A}

collec-tionβ fii∈J of openL-fuzzy sets is called ap-level open cover of theL-fuzzy setgif and

only if_i∈Jfixpfor allx∈Xwithgx≥p.

Ifgis the whole spaceX1, thenβis called ap-level open cover ofX.

Thus β is called a p-level open cover of X if and only if _f∈βfx p for all

x∈X.

Definition 2.9see4. LetX, τbe anL-fuzzy topological space andg ∈LX_{. TheL-fuzzy set}

gis said to be fuzzy compact if and only if everyp-level open cover ofg, wherep∈prL, has a finitep-level subcover ofg.

If g is the whole space X, then X, τ is called fuzzy compact L-fuzzy topological space.

3. Proposed definitions

Definition 3.1. LetXbe nonempty set and letβ, γ ⊆LX_{.βis called a partial inversement ofγ if}

and only ifβandγcan be indexed with the same index set, sayJ:β{fi:i∈J},γ {gi:i∈J}

so that for everyi∈Jeitherfigorfig_i.

Definition 3.2. Let X, τbe anL-fuzzy topological space andg ∈ LX_{.g is said to be fuzzy}

inversely compact if and only if everyp-level open cover ofg, wherep∈prL, has a partial inversement which contains a finitep-level cover ofg.

Ifgis the whole spaceX, thenX, τis called fuzzy inverse compactL-fuzzy topological space.

This definition can be restated as follows.gis fuzzy inverse compact if and only if for everyp-level open coverβ {fi : i ∈ J}ofg, wherep ∈ prL, there exists a finitep-level

subcoverγ {gi :i ∈J}ofgsuch that there existg1, g2, . . . , gn ∈γ :ni1gixpfor every

x∈Xwithgx≥p, where for eachi1,2, . . . , n,gifi, orgifi.

Clearly, every fuzzy compactL-fuzzy set is inversely fuzzy compact.

Definition 3.3. Let X, τbe anL-fuzzy topological space andg ∈ LX_{.g is said to be fuzzy}

inversely countably compact if and only if for every countablep-level open cover ofg, where

p∈prL, has a partial inversement which contains a finitep-level cover ofg.

Ifg is the whole space, then we say that theL-fuzzy topological spaceX, τis fuzzy inversely countably compact.

Definition 3.4. Let X, τbe anL-fuzzy topological space andg ∈ LX_{.g is said to be fuzzy}

inversely Lindel ¨of if and only if everyp-level open cover ofg, wherep∈prL, has a partial inversement which contains a countablep-level cover ofg.

From the definitions, it can be easily verified that every fuzzy inversely compactL-fuzzy setgis fuzzy inversely countably compact and fuzzy inversely Lindel ¨of.

4. Other characterizations

Definition 4.1. Letα∈L,ζ⊆LX_{, andg ∈L}X_{.ζis called anα-level centered family ofgif and}

only if for anyh1, h2, . . . , hn∈ζthere existsx∈Xwithgx≥αsuch thath1∧h2∧· · ·∧hnx≥α.

Theorem 4.2. LetX, τbe anL-fuzzy topological space andg ∈LX_{.gis fuzzy compact if and only} if for everyα-level centered familyζof closedL-fuzzy sets, whereα∈ML, there existsx∈Xwith

gx≥αsuch that_h∈ζhx≥α.

Proof

Necessity. Letα∈MLand letζbeα-level centered family of closedL-fuzzy sets such that for eachx∈Xwithgx≥α, there existsh∈ζwithhxα.

Then,_h∈ζhxαfor allx∈Xwithgx≥α.

Hence_h∈ζhxpfor allx∈Xwithgx≥p, wherepα.

Thus,β{h :h∈ζ}is ap-level open cover ofgthat has no finitep-level subcover of

g. In fact, ifh₁, h₂, . . . , h_n∈β, then, sinceζisα-level centered, there existsx∈Xwithgx≥α

andn_i1hix≥α; henceni1hix≤p.

Sufficiency. Suppose that βis a p-level open cover ofg with no finitep-level subcover ofg, wherep∈prL. Then,ζ{f:f∈β}is a collection of closedL-fuzzy sets. Moreover,ζis an

α-level centered family ofg, whereα p. In fact, iff₁, f₂, . . . , f_n ∈ζ, then there existsx∈X

withgx≥pαsuch thatn_i1fix≤p; henceni1fix≥α.

By the hypothesis, there existsx ∈ X withgx ≥ pwith such that_f∈βfx ≥ p; hence_f∈βfx≤pwhich yields a contradiction.

Definition 4.3. Letζ⊆LX_,g∈LX_{, andα∈L.}

ζis called anα-level independent family ofg if and only if for any finitef1, f2, . . . , fn,

g₁, g2, . . . , gm∈ζthere existsx∈Xwithgx≥αsuch thatni1fi∧mi1gix≥α.

In other words,ζis anα-level independent family ofgif and only if for every nonempty finite partial inversementζ∗ofζ, there existsx∈Xwithgx≥αand_h∈ζ∗hx≥α.

Theorem 4.4. LetX, τbe anL-fuzzy topological space andg ∈LX_{.gis fuzzy inversely compact if} and only if for everyα-level independent familyζof closedL-fuzzy sets, whereα∈ML, there exists x∈Xwithgx≥αand_h∈ζhx≥α.

Proof

Necessity. Letα∈ML. Suppose thatζis anα-level independent family ofL-fuzzy sets, such that_h∈ζhxαfor everyx∈Xwithgx≥α.

Then,_h∈ζhxαfor allx∈Xwithgx≥α.

Hence,β{h:h∈ζ}is ap-level open cover ofg, wherepα.

By the fuzzy inverse compactness ofg, there is a partial inversementγ ofβwhich con-tains a finitep-level subcover ofg.

Letγ {gi:i∈J}andβ{h_i:i∈J,hi∈ζ}.

Then, there existsg1, g2, . . . , gn∈γsuch that

_n

i1gixpfor allx∈Xwithgx≥p,

Letg1 h₁, g2 h₂, . . . , gk h_k, gk1 hk1, gk2 hk2, . . . , gn hn. Then g₁, g₂,

. . . , g_k, gk1, . . . , gn ∈ζ. Sinceζis anα-level independent family ofg, there existsx ∈Xwith

gx≥αsuch that

k

i1 g_i∧

n

ik1 g_i

x n

i1 g_i

x≥α. 4.1

Hence, there existsx ∈ X with gx ≥ p such that n_i1gi

x ≤ p, which yields a contradiction.

Sufficiency. Suppose thatβis ap-level open cover ofgwith no partial inversement which con-tains a finitep-level subcover ofg, wherep ∈prL. Then,ζ {f : f ∈β}is a collection of closedL-fuzzy sets. Furthermore,ζis anα-level independent family ofg, whereαp. In fact, iff₁, f₂, . . . , f_k, f_k1, . . . , f_n ∈ζ,then by the assumption, there existsx∈Xwithgx≥pand

k

i1fi∨nik1fix≤p; hence

k i1fi∧

n

ik1fix≥α.

By the hypothesis, there exists z ∈ X with gz ≥ α and _f∈βfz ≥ α; hence

f∈βfz≤p,which yields a contradiction.

Theorem 4.5. LetX, τbe anL-fuzzy topological space andg ∈LX_{.gis fuzzy inversely countably} compact if and only if for every countableα-level independent familyζof closedL-fuzzy sets, where

α∈ML, there existsx∈Xwithgx≥αand_h∈βhx≥α.

Proof. This is similar toTheorem 4.4.

Definition 4.6. Letξ ⊆LX_{andp ∈ prL.ξis said to have the finite union propertyfor short}

FUPinG{x∈X:gx≥p}if and only if for any finitef1, f2, . . . , fk, g1, g2, . . . , gn∈ξ,there

existsx∈Gwithk_i1fi∨

_n

i1gix≤p.

Theorem 4.7. LetX, τbe anL-fuzzy topological space andg ∈LX_{.gis fuzzy inversely compact if} and only if noξ⊆τwith the FUP inGsatisfies_f∈ξfxpfor allx∈G.

Proof

Necessity. Suppose thatξ⊆τhas the FUP inGsuch that_f∈ξfxpfor allx∈G.

Then,ζ {f : f ∈ ξ}is anα-level independent family of closedL-fuzzy sets, where

α p. From Theorem 4.4, there existsx ∈ X with gx ≥ αand _f∈ζfx ≥ α; hence

f∈ξfx≤pwhich yields a contradiction.

Sufficiency. Suppose thatgis not fuzzy inverse compact. Then, there is anα-level independent familyζof closedL-fuzzy sets such that_h∈ζhx αfor allx∈ X withgx ≥α,where

α ∈ ML. Hence_h∈ζhx pfor all x ∈ X withgx ≥ p, wherep α. Moreover,

ξ {h : h ∈ ζ} is a family of openL-fuzzy sets having the FUP inG. This completes the proof.

5. Some properties

The next theorem shows that fuzzy inverse compactness is a good extension of inverse com-pactness in general topology.

Theorem 5.1. LetX, Tbe an ordinary topological space.X, Tis inversely compact if and only if the

Proof

Sufficiency. Letα∈MLandζbe anα-level independent family of closed subsets ofX. Then,

ζ∗{χA:A∈ζ}is a family of closedL-fuzzy sets. Furthermore,ζ∗is anα-level independent

family ofX. In fact, ifχA1, χA2, . . . , χAn, χB1,...,χBm ∈ ζ∗,then sinceζis independent, n

i1Ai∩

_m

i1Bi/φ. Hence

_n

i1χAi∧ _m

i1χBi χ

n i1Ai∩

m

i1Bi/0. So, there isx∈Xsuch that _n

i1χAi∧ m

i1χBix 1 ≥ α. SinceX, ωLTis fuzzy inversely compact, there exists z ∈ X with

_A∈ζχAz≥α; henceA∈ζχAz χA∈ζAz 1 and thereforez∈

A∈ζA. HenceX, T

is inversely compact.

Necessity. Letp ∈prLandβ fii∈J be ap-level open cover ofXconsisting of basic open

L-fuzzy sets inX, ωLT. Then

fix

⎧ ⎨ ⎩

ei∈L; x∈Ai∈T,

0; x /∈Ai

5.1

for eachi∈Jand_i∈Jfixpfor allx∈X.

Letζ{Ai⊆X:∃i∈Jwitheipandfi∈β}.

It is clear thatζis an open cover ofX inX, T. Due to inverse compactness ofX, T, there exists a partial inversementζ∗ofζwhich contains a finite cover ofX.

Letζ∗ {Bi : i ∈ J}. Then there existB1, B2, . . . , Bn ∈ ζ∗ such thatX ni1Bi,where

BiAiandBiA_ifor eachi∈ {1,2, . . . , n}.

iIfBi Ai for alli ∈ {1,2, . . . , n}, thenβ∗ {f1, f2, . . . , fn}is a finite partial

inverse-ment ofβ.β∗is also ap-level subcover ofXinX, ωLT. Hence,X, ωLTis fuzzy

inversely compact.

iiSuppose that there existsi0∈ {1,2, . . . , n}such thatBi0Ai0.

Let β∗ {f1, f2, . . . , fi0−1, f

i0, . . . , f

n}. Obviously, β∗ is a partial inversement of β.

Fur-thermore,β∗is ap-level subcover ofX inX, ωLT. In fact, letx ∈ X. SinceX ni1Bi

Bi0∪

n

i1,i /i0Bi⇒x∈Bi0orx∈

n

i1,i /i0Bi⇒x∈A

i0orx∈

n

i1,i /i0Ai.

1◦Ifx∈A_i

0, thenf

i0x 1p. Hencef

i0x∨

_n

i1,i /i0fixp.

2◦ Ifx /∈ A_i

0, then there isi ∈ {1,2, . . . , n} − {i0} such thatx ∈ Ai. Hence fix

ei pand thereforefi0∨

_n

i1,i /i0fix p.Consequently, β

∗ _{isp-level subcover ofX in} X, ωLT.

Theorem 5.2the goodness of fuzzy inverse countable compactness. LetX, Tbe an ordinary topological space. X, Tis inversely countably compact if and only if the L-fuzzy topological space X, ωLTis fuzzy inversely countably compact.

Proof. This is similar to the proof of the goodness of fuzzy inverse compactness.

Theorem 5.3the goodness of fuzzy inverse Lindel ¨ofness. LetX, Tbe an ordinary topological space.X, Tis an inverse Lindel¨of space if and only if theL-fuzzy topological spaceX, ωLTis fuzzy inverse Lindel¨of space.

Proof. This is similar to the proof of the goodness of fuzzy inverse compactness.

Theorem 5.4. LetX, τbe anL-fuzzy topological space. Ifg ∈ LX _{is fuzzy inversely compact and}

Proof. Letp∈prLand letβ fii∈J⊆LXbe ap-level open cover ofg∧h; that is,

i∈Jfix

pfor allx∈Xwithg∧hx≥p. Soβ∗ fii∈J∪ {h}isp-level family ofg. In fact, if

hx≥p⇒

i∈J

fi

xp ∀x∈X withg∧hx≥p

⇒

k∈β∗

k

x

i∈J

fi

x∨hxp

hxp⇒hxp⇒

k∈β∗ k

xp.

5.2

Since g is fuzzy inversely compact, β∗ has a partial inversement γ∗ which contains a finite

p-level subcover ofg.

Letγ∗{gi:i∈J} ∪ {h}andβ∗{fi:i∈J} ∪ {h}.

Then there existsg1, g2, . . . , gn ∈ γ∗such that ni1gi∨hx p for allx ∈ X with

gx≥pwhere for eachi∈ {1,2, . . . , n}, gifiorgifi.

Son_i1gixpfor allx∈Xwithg∧hx≥p. In factx∈Xwithg∧hx≥p⇒

gx≥pandhx≥p,

gx≥p⇒ n

i1 gi∨h

xp

⇒n

i1 gi

x∨hxp

⇒

n

i1 gi

xp.

5.3

Corollary 5.5. LetX, τbe anL-fuzzy topological space. Ifgis a fuzzy inversely compactL-fuzzy set andhis a closedL-fuzzy set withh≤g, thenhis fuzzy inversely compact as well.

Proof. This follows directly from the previous theorem.

Corollary 5.6. LetX, τbe anL-fuzzy topological space. IfX, τis fuzzy inversely compact, then every closedL-fuzzy set onXis fuzzy inversely compact.

Proof. This follows fromTheorem 5.4.

Proposition 5.7. LetX, τbe anL-fuzzy topological space andg, h∈LX_{withh≤g. Ifgis fuzzy} in-versely countably compact (fuzzy inin-versely Lindel¨of) andhis closed, thenhis fuzzy inversely countably compact (fuzzy inversely Lindel¨of).

Proof. This is similar toTheorem 5.4.

Proof. Suppose thatg∨his not fuzzy inverse compact.

Then, there exists anα-level independent familyζofg∨hconsisting of closedL-fuzzy sets such that_f∈ζfx≥αfor allx∈Xwithg∨hx≥α, whereα∈ML.

Sinceα∈MLwe haveg∨hx≥α⇔gx≥αorhx≥α.

Letf1, f2, . . . , fk, fk1, . . . , fn∈ζ. Sinceζis anα-level independent family ofg∨h, there

existsx∈Xwithg∨hx≥αandk_i1fi∧nik1fix≥α. That is, there existsx∈Xwith

gx≥αorhx≥αsuch thatk_i1fi∧

_n

ik1fix≥α.

Assume thatgx ≥ α. Then,ζisα-level independent family ofg consisting of closed

L-fuzzy sets. Furthermore,_f∈ζfxαfor allx∈Xwithgx≥α.

So,gis not fuzzy inversely compact.

Proposition 5.9. Let X, τ be an L-fuzzy topological space and g, h ∈ LX_{. If g and h are fuzzy} inversely countably compact (fuzzy inversely Lindel¨of), theng∨his fuzzy inversely countably compact (fuzzy inversely Lindel¨of).

Proof. This is similar toTheorem 5.8.

Theorem 5.10. LetX, τandY, τ∗beL-fuzzy topological spaces and letϕ:X, τ→Y, τ∗be a fuzzy continuous function such thatϕ−1_{yis finite for everyy∈Y. Ifg ∈L}X_{is fuzzy inversely} com-pact inX, τ, thenϕg∈LY _{is fuzzy inversely compact inY, τ}∗_{, whereϕgy}

x∈ϕ−1_ygx.

Proof. Letα∈MLandζis anα-level independent family ofϕgconsisting of closedL-fuzzy sets inY, τ∗. Letζ∗{ϕ−1_h:h∈ζ}.

Since ϕ is fuzzy continuous, ζ∗ is a family of closed L-fuzzy sets in X, τ. Further-more, ζ∗ is an α-level independent family of g. In fact, if ϕ−1_{h1, ϕ}−1_{h2, . . . , ϕ}−1_h

k,

ϕ−1_h

k1, . . . , ϕ−1hn∈ζ∗, then there existsy∈Ywithϕgy≥αandki1hi∧nik1hiy≥

αbecauseζ is anα−level independent family ofϕg.ϕgy _x∈ϕ−1_ygx ≥ αimplies

that there existsx ∈ Xwithgx ≥ αandϕx ybecause α ∈ MLandϕ−1_{yis finite.}

Hence,k_i1ϕ−1_h

i∧nik1ϕ−1hix

k

i1hi ∧nik1hiϕx ≥ α. From the fuzzy

in-verse compactness ofg, there exists z ∈ X with gz ≥ αand _h∈ζϕ−1_{hz ≥ α. Then,}

h∈ζhϕz≥α. This means thatϕgis fuzzy inversely compact inY, τ∗.

Theorem 5.11. LetX, τandY, τ∗beL-fuzzy topological spaces and letϕ : X, τ → Y, τ∗be a fuzzy continuous function such thatϕ−1_{yis finite for everyy ∈ Y. Ifg ∈L}X _{is fuzzy inversely} countable compact (fuzzy inversely Lindel¨of) inX, τ, thenϕg ∈LY _{is fuzzy inversely countably} compact (fuzzy inversely Lindel¨of) inY, τ∗.

Proof. This is similar to the proof ofTheorem 5.10.

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