almost contra continuity in topological spaces

Popae a Departamento de Matematica Aplicada, Universidade Federal Fluminense, Rua Mario Santos Braga, s/n 24020-140, Niteroi, RJ, Brazil b Department of Mathematics, Graz University of

journalhomepage:www.elsevier.com/locate/joems

Original Article

M Caldasa, M Gansterb, S Jafaric,∗, T Noirid, V Popae

a Departamento de Matematica Aplicada, Universidade Federal Fluminense, Rua Mario Santos Braga, s/n 24020-140, Niteroi, RJ, Brazil

b Department of Mathematics, Graz University of Technology, Steyrergasse 30, A-8010 Graz, Austria

c College of Vestsjaelland South, Herrestraede 11, 4200 Slagelse, Denmark

d 2949-1 Shiokita-cho, Hinagu, Yatsushiro-shi, Kumamoto-ken, 869-5142, Japan

e Department of Mathematics and Informatics, Faculty of Sciences “Vasile Alecsandri”, University of Bac ˇau, 157 Calea M ˇar ˇa ¸s e ¸s ti, Bac ˇau, 600115, Romania

Article history:

Received 27 April 2016

Revised 16 August 2016

Accepted 20 August 2016

Available online xxx

MSC:

Primary 54C08

54C10

Secondary 54C05

keywords:

Topological space

βθ-Open set

βθ-Closed set

Almost contra-continuous function

Almost contra βθ-continuous function

Inthispaper,weintroduceandinvestigatethenotionofalmostcontraβθ-continuousfunctionsby uti-lizingβθ-closedsets.Weobtainfundamentalpropertiesofalmostcontraβθ-continuousfunctionsand discusstherelationshipsbetweenalmostcontraβθ-continuityandotherrelatedfunctions

© 2016EgyptianMathematicalSociety.ProductionandhostingbyElsevierB.V

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1 Introduction and preliminaries

Recently, Baker (resp Ekici, Noiri and Popa) introduced and

investigated the notions of contra almost β-continuity [1](resp

almost contra pre-continuity [2,3]) as a continuation of research

done by Caldas and Jafari [4] (resp Jafari and Noiri [5]) on the

notionofcontra-β-continuity(resp.contrapre-continuity) Inthis

paper, new generalizations of contra βθ-continuity [6] by using

βθ-closed sets calledalmost contra βθ-continuityare presented

Weobtainsomecharacterizationsofalmostcontraβθ-continuous

functionsandinvestigatetheirpropertiesandtherelationships

be-tween almost contra βθ-continuityandother relatedgeneralized

formsofcontinuity

Throughoutthispaper,by(X,τ)and(Y,σ)(orXandY)we

al-waysmeantopologicalspaces LetA be asubset ofX.Wedenote

theinterior,the closureandthecomplementofaset Aby Int(A),

R Dedicated to our friend and colleague the late Professor Mohamad Ezat Abd El-

Monsef

∗ Corresponding author

E-mail addresses: gmamccs@vm.uff.br (M Caldas), ganster@weyl.math.tu-

graz.ac.at (M Ganster), jafaripersia@gmail.com (S Jafari), t.noiri@nifty.com (T Noiri),

v.popa@ub.ro (V Popa)

Cl(A) and X\A,respectively Asubset A of X is said to be regular open(resp.regularclosed)ifA=Int(Cl(A))(resp.A=Cl(Int(A)))

A subset A of a space X is called preopen [7] (resp semi-open

[8], β-open [9], α-open [10]) if A ⊂ Int(Cl(A)) (resp A ⊂ Cl(Int(A)),

A ⊂ Cl(Int(Cl(A))),A ⊂ Int(Cl(Int(A)))).Thecomplementofa preopen (resp semi-open, β-open, α-open) set is said to be preclosed (resp.semi-closed,β-closed,α-closed) The collectionofall open (resp closed, regular open, preopen, semiopen, β-open) subsets

of X will be denoted by O(X) (resp C(X), RO(X), PO(X), SO(X),

βO(X)) We set RO(X , x)={U:x∈U∈RO(X ,τ ) }, SO(X , x)={U:

x∈U∈SO(X ,τ ) }andβO(X , x)={U:x∈U∈βO(X ,τ ) }.Wedenote thecollection ofall regular closed subsetsof X by RC(X) We set

RC(X , x)={U:x∈U∈RC(X ,τ ) }.Wedenotethecollectionofallβ -regular (i.e., if it is both β-open and β-closed) subsets of X by

βR(X) Apoint x ∈X is said tobe a θ-semi-cluster point [11] of

asubset Sof XifCl(U)∩A = ∅forevery U∈SO(X, x) Theset of allθ-semi-clusterpointsofAiscalledtheθ-semi-closureofAand

is denoted by θsCl(A) A subset A is called θ-semi-closed [11] if

A=θsCl(A) The complement of a θ-semi-closed set is called

θ-semi-open

Theβθ-closureofA [12],denotedbyβCl θ(A),is definedto be thesetof allx ∈X such that βCl(O)∩A =∅forevery O ∈βO(X,

τ)withx∈O.Theset x∈X:βCl θ(O)⊂ A for some O∈βO(X, x)} http://dx.doi.org/10.1016/j.joems.2016.08.002

1110-256X/© 2016 Egyptian Mathematical Society Production and hosting by Elsevier B.V This is an open access article under the CC BY-NC-ND license

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iscalledtheβθ-interiorofAandisdenotedbyβInt θ(A).Asubset

Aissaid tobeβθ-closed[12]ifA=βCl θ(A).Thecomplementof

aβθ-closedsetissaid tobeβθ-open Thefamilyofallβθ-open

(resp.βθ-closed)subsetsofXisdenotedbyβθO(X,τ) orβθO(X)

(resp.βθC(X,τ)).WesetβθO(X , x)={U:x∈U∈βθO(X ,τ ) }and

βθC(X , x)={U:x∈U∈βθC(X ,τ ) }

We recall the following two lemmas whichwere obtainedby

Noiri[12]

Lemma 1.1[12]. Let A be a subset of a topological space(X,τ).

(i) If A∈βO(X,τ),thenβCl(A)∈βR(X).

(ii) A∈βR(X)if and only if A∈βθO(X)∩βθC(X).

Lemma 1.2 [12]. For theβθ-closure of a subset A of a topological

space(X,τ),the following properties are hold:

(i) A⊂βCl(A)⊂βCl θ(A)andβCl(A)=βCl θ(A)if A∈βO(X).

(ii) If A ⊂ B, thenβCl θ(A)⊂βCl θ(B).

(iii)If A α ∈βθC(X)for eachα ∈A, then{A α| α∈A}∈βθC(X).

(iv)If A α ∈βθO(X)for eachα∈A, then{A α| α∈A}∈βθO(X).

(v) βCl θ( βCl θ(A))=βCl θ(A)andβCl θ(A)∈βθC(X).

Definition 1. Afunctionf:X→Yissaidtobe:

(1)βθ-continuous [12]if f−1(V) isβθ-closedforevery closed

set V in Y, equivalentlyifthe inverseimage ofevery open

setVinYisβθ-openinX

(2)Almostβθ-continuousif f−1(V)isβθ-closedinXforevery

regularclosedsetVinY

(3)ContraR-maps[13](resp.contra-continuous[14],contraβθ

-continuous[6])iff−1(V)isregularclosed(resp.closed,βθ

-closed) inXforevery regularopen(resp.open,open)setV

ofY

(4)Almost contra pre-continuous [2] (resp almost contra β

-continuous [1], almost contra -continuous [1]) if f−1(V) is

preclosed(resp.β-closed,closed)inXforeveryregularopen

setVofY

(5)Regularset-connected[15]iff−1(V)isclopeninXforevery

regularopensetVinY

2 Characterizations

Definition 2. Afunctionf:X→Yissaidtobealmostcontraβθ

-continuousiff−1(V)isβθ-closedinXforeachregularopensetV

ofY

Definition 3. LetAbe asubset of a space(X,τ) The set {U∈

RO(X):A ⊂ U} is calledthe r-kernelof A [13] andis denotedby

rker(A)

Lemma 2.1(Ekici[13]). For subsets A and B of a space X, the

follow-ing properties hold:

(1)x∈rker(A)if and only if A∩F=∅for any F∈RC(X, x).

(2)A ⊂ rker(A)and A= ker(A)if A is regular open in X.

(3)If A ⊂ B, then rker(A)⊂ rker(B)

Theorem 2.2. For a function f: X→ Y, the following properties are

equivalent:

(1)f is almost contraβθ-continuous;

(2)The inverse image of each regular closed set in Y isβθ-open in

X;

(3)For each point x in X and each V∈RC(Y, f(x)),there is a U∈

βθO(X, x)such that f(U)⊂ V;

(4)For each point x in X and each V∈SO(Y, f(x)),there is a U∈

βθO(X, x)such that f(U)⊂ Cl(V);

(5)f(βCl θ(A))⊂ rker(f(A))for every subset A of X;

(6)βCl θ(f−1(B))⊂ f−1(rker(B))for every subset B of Y;

(7) f−1(Cl(V))isβθ-open for every V∈βO(Y);

(8) f−1(Cl(V))isβθ-open for every V∈SO(Y);

(9) f−1(Int(Cl(V)))isβθ-closed for every V∈PO(Y);

(10) f−1(Int(Cl(V)))isβθ-closed for every V∈O(Y);

(11) f−1(Cl(Int(V)))isβθ-open for every V∈C(Y).

Proof. (1)⇔(2):seeDefinition2 (2)⇔(4): Let x ∈ X and V be anysemiopen set of Y contain-ingf(x),thenCl(V)isregularclosed.By(2) f−1(Cl(V))isβθ-open andthereforethereexistsU∈βθO(X, x)suchthatU ⊂ f−1(Cl(V)) Hencef(U)⊂ Cl(V)

Conversely,supposethat (4)holds.LetVbeanyregularclosed setofY andx∈f−1(V).Then V isa semiopensetcontainingf(x) andthereexistsU∈βθO(X, x)suchthatU ⊂ f−1(Cl(V))= f−1(V) Therefore, x∈U ⊂ f−1(V) andhence x∈U⊂βInt θ(f−1(V)) Con-sequently, we have f−1(V)⊂βInt θ(f−1(V)) Therefore f−1(V)=

βInt θ(f−1(V)), i.e., f−1(V)isβθ-open

(2)⇒(3):Letx∈XandVbearegularclosedsetofYcontaining

f(x).Thenx∈f−1(V).Sincebyhypothesis f−1(V)isβθ-open,there exists U ∈βθO(X, x) such that x∈U ⊂ f−1(V) Hence x ∈U and

f(U)⊂ V (3)⇒(5): Let A be anysubset ofX.Suppose that y∈rker(f(A)) Then,byLemma2.1thereexistsV∈RC(Y, y)suchthat f(A)∩V=

∅ For any x∈f−1(V), by (3) there exists U x ∈ βθO(X, x) such that f(U x)⊂ V Hence f(A∩U x)⊂ f(A)∩f(U x)⊂ f(A)∩V=∅ and

A∩U x=∅.This showsthat x∈βCl θ(A) for anyx∈f−1(V) There-fore, f−1(V)∩βCl θ(A)=∅ and hence V∩f( βCl θ(A))=∅ Thus,

y∈f(βCl θ(A)).Consequently,weobtainf(βCl θ(A))⊂ rker(f(A)) (5)⇔(6): Let B be any subset of Y By (5) and Lemma 2.1,

we have f( βCl θ(f−1(B)))⊂ r ker(f f−1(B))⊂ r ker(B) and

βCl θ(f−1(B))⊂ f−1(rker(B)) Conversely, suppose that (6) holds Let B=f(A), where A is

a subset of X Then βCl θ(A)⊂βCl θ(f−1(B))⊂ f−1(rker(f(A))) Thereforef(βCl θ(A))⊂ rker(f(A))

(6)⇒(1): LetV beanyregular open set ofY.Then, by(6) and

Lemma2.1(2)wehaveβCl θ(f−1(V))⊂ f−1(rker(V))= f−1(V)and

βCl θ(f−1(V))=f−1(V).Thisshowsthat f−1(V)isβθ-closedinX Thereforefisalmostcontraβθ-continuous

(2)⇒(7): Let V be anyβ-open set of Y It follows from([16], Theorem 2.4)that Cl(V) isregular closed.Then by (2) f−1(Cl(V))

isβθ-openinX (7)⇒(8): This is clear since every semiopen set is β-open (8)⇒(9): Let V be any preopen set of Y Then Int(Cl(V)) is regu-lar open Therefore Y\Int(Cl(V)) is regular closed and hence it is semiopen Then by (8) X\f−1(Int(Cl(V)))= f−1(Y\Int(Cl(V)))=

f−1(Cl(Y\Int(Cl(V)))) is βθ-open Hence f−1(Int(Cl(V))) is βθ -closed

(9)⇒(1):LetVbeanyregularopensetofY.ThenVispreopen andby(9) f−1(V)= f−1(Int(Cl(V)))isβθ-closed.Itshowsthat f

isalmostcontraβθ-continuous

(1)⇔(10):LetVbeanopensubsetofY.SinceInt(Cl(V))is regu-laropen, f−1(Int(Cl(V)))isβθ-closed.Theconverseissimilar (2)⇔(11):Similarto(1)⇔(10) 

Lemma 2.3 [17]. For a subset A of a topological space (Y, σ), the following properties hold:

(1) αCl(A)=Cl(A)for every A∈βO(Y).

(2) pCl(A)=Cl(A)for every A∈SO(Y).

(3) sCl(A)=Int(Cl(A))for every A∈PO(Y).

Corollary 2.4. For a function f: X→ Y, the following properties are equivalent:

(1) f is almost contraβθ-continuous;

(2) f−1( αCl(A))isβθ-open for every A∈βO(Y);

(3) f−1(pCl(A))isβθ-open for every A∈SO(Y);

(4) f−1(sCl(A)))isβθ-closed for every A∈PO(Y).

Proof. ItfollowsfromLemma2.3 

Theorem 2.5. For a function f:X →Y, the following properties are

equivalent:

(1) f is almost contraβθ-continuous;

(2) the inverse image of aθ-semi-open set of Y isβθ-open;

(3) the inverse image of aθ-semi-closed set of Y isβθ-closed;

(4) f−1(V)⊂βInt θ(f−1(Cl(V)))for every V∈SO(Y);

(5) f(βCl θ(A))⊂θsCl(f(A))for every subset A of X;

(6) βCl θ(f−1(B))⊂ f−1( θsCl(B))for every subset B of Y;

(7) βCl θ(f−1(V))⊂ f−1( θsCl(V))for every open subset V of Y;

(8) βCl θ(f−1(V))⊂ f−1(sCl(V))for every open subset V of Y;

(9) βCl θ(f−1(V))⊂ f−1(Int(Cl(V)))for every open subset V of Y.

Proof. (1)⇒(2): Since any θ-semiopen set is a union of regular

closedsets,byusing(1)andTheorem2.2,weobtainthat(2)holds

(2)⇒(1):Letx∈X andV∈SO(Y) containingf(x).SinceCl(V) is

θ-semiopenin Y,there exists a βθ-open set U inX containingx

suchthatx∈U ⊂ f−1(Cl(V)).Hencef(U)⊂ Cl(V)

(1)⇒(4):LetV∈SO(Y)andx∈f−1(V).Thenf(x)∈V.By(1)and

Theorem2.2,thereexistsaU∈βθO(X, x)suchthatf(U)⊂ Cl(V).It

followsthatx∈U ⊂ f−1(Cl(V)).Hencex∈βInt θ(f−1(Cl(V))).Thus

f−1(V)⊂βInt θ(f−1(Cl(V)))

(4)⇒(1): LetFbe anyregularclosed setofY.Since F∈SO(Y),

then by (4), f−1(F)⊂βInt θ(f−1(F)). This shows that f−1(F) is

βθ-open,byTheorem2.2,(1)holds

(2)⇔(3):Obvious

(1)⇒(5):LetAbeanysubsetofX.Supposethatx∈βCl θ(A)and

GisanysemiopensetofYcontainingf(x).By(1)andTheorem2.2,

thereexistsU∈βθO(X, x)suchthatf(U)⊂ Cl(G).Sincex∈βCl θ(A),

U∩A=∅andhence∅=f(U)∩f(A)⊂ Cl(G)∩f(A).Therefore,we

ob-tainf(x)∈θsCl(f(A))anhencef(βCl θ(A))⊂θsCl(f(A))

(5)⇒(6): Let B be any subset of Y Then f( βCl θ(f−1(B)))⊂

θsCl(f(f−1(B))⊂θsCl(B)andβCl θ(f−1(B))⊂ f−1( θsCl(f(B)).

(6)⇒(1): Let V be any semiopen set of Y containing f(x)

Since Cl(V)∩(Y\Cl(V))=∅ we have f(x)∈θsCl(Y\ClV)) and x∈/

f−1( θsCl(Y\Cl(V))).By (6), x∈/βCl θ(f−1(Y\Cl(V))).Hence, there

exists U ∈βθO(X, x) such that U∩f−1(Y\Cl(V))=∅and f(U)∩

(Y\Cl(V))=∅.Itfollowsthatf(U)⊂ Cl(V).Thus,byTheorem2.2,we

havethat(1)holds

(6)⇒(7):Obvious

(7)⇒(8): Obvious from the fact that θsCl(V)=sCl(V) for an

opensetV

(8)⇒(9):ObviousfromLemma2.3

(9)⇒(1): Let V ∈ RO(Y) Then by (9) βCl θ(f−1(V))⊂

f−1(Int(Cl(V)))=f−1(V). Hence, f−1(V) is βθ-closed which

provesthatfisalmostcontraβθ-continuous 

Corollary 2.6. For a function f:X →Y, the following properties are

equivalent:

(1) f is almost contraβθ-continuous;

(2) βCl θ(f−1(B))⊂ f−1( θsCl(B))for every B∈SO(Y).

(3) βCl θ(f−1(B))⊂ f−1( θsCl(B))for every B∈PO(Y).

(4) βCl θ(f−1(B))⊂ f−1( θsCl(B))for every B∈βO(Y).

Proof. In Theorem 2.5, we have proved that the following are

equivalent:

(1) fisalmostcontraβθ-continuous;

(2) βCl θ(f−1(B))⊂ f−1( θsCl(B))foreverysubsetBofY

Hencethecorollaryisproved 

Recall that a topological space(X, τ) is said to be extremally

disconnectediftheclosureofeveryopensetofXisopeninX

Theorem 2.7. If(Y,σ)is extremally disconnected, then the following

properties are equivalent for a function f:X→Y:

(1) f is almost contraβθ-continuous;

(2) f is almostβθ-continuous.

Proof. (1)⇒(2):Letx∈XandUbeanyregularopensetofY con-tainingf(x) Since Yis extremallydisconnected, by Lemma5.6of

[18]UisclopenandhenceUisregularclosed.Then f−1(U)isβθ -openinX.Thusfisalmostβθ-continuous

(2)⇒(1): Let B be any regular closed set of Y Since Y is ex-tremallydisconnected,Bisregularopenand f−1(B)isβθ-openin

X.Thusfisalmostcontraβθ-continuous

Thefollowingimplicationsareholdforafunctionf:X→Y:

F

Notation: A= almost contra β-continuity, B= almost con-tra βθ-continuity, C= contra βθ-continuity, D= almost contra-continuity, E= almost contra pre-continuity, F= contra R-map,

G=contraβ-continuity,H=almostcontrasemi-continuity 

Example 2.8. Let (X, τ) be a topological space such that

X={a , b , } and τ={∅,{b},{ },{b , }, X}. Clearly βθO(X ,τ )= {∅,{b},{ },{a , b},{a , },{b , }, X}.Letf:X→Xbedefinedby f(a)=

c , f(b)=b and f(c)=a Then f is almost contra βθ-continuous butfisnot contraβθ-continuous, notβθ-continuous andalsois notcontracontinuous

Otherimplicationsnotreversibleareshownin[2,3,5,6,13,15]

Theorem 2.9. If f:X→ Y is an almost contraβθ-continuous func-tion which satisfies the property βIn t θ((f−1(Cl(V))))⊂ f−1(V) for each open set V of Y, then f isβθ-continuous.

Proof. Let V be any open set of Y Since f is almost con-tra βθ-continuous by Theorem 2.2 f−1(V)⊂ f−1(Cl(V))=

βIn t θ( βIn t θ(f−1(Cl(V))))⊂βIn t θ(f−1(V))⊂ f−1(V) Hence

f−1(V)isβθ-openandthereforefisβθ-continuous

Recall that atopological spaceis saidto be P  [19]ifforany opensetVofXandeachx∈V,thereexistsaregularclosedsetF

ofXcontainingxsuchthatx∈F ⊂ V 

Theorem 2.10. If f:X→Y is an almost contraβθ-continuous func-tion and Y is P  , then f isβθ-continuous.

Proof. SupposethatVisanyopensetofY.BythefactthatYisP  ,

sothereexistsasubfamilyofregularclosedsetsofYsuchthat

V={F|F∈ } Since f is almost contra βθ-continuous, then

f−1(F) isβθ-open inX foreach F∈.Therefore f−1(V) isβθ -openinX.Hencefisβθ-continuous

Recallthatafunctionf:X→Yissaidtobe:

a)R-map [20] (resp pre βθ-closed [21]) if f−1(V) is regular closed in X forevery regular closed V of Y(resp f(V) is βθ -closedinYforeveryβθ-closedVofX)

b) weakly β-irresolute [12] if f−1(V) is βθ-open in X forevery

βθ-opensetVinY 

Theorem 2.11. Let f:X → Y and g: Y→ Z be functions Then the following properties hold:

(1) If f is almost contra-βθ-continuous and g is an R-map, then

g ◦ f:X→Z is almost contraβθ-continuous.

(2) If f is almost βθ-continuous and g is a contra R-map, then

g ◦ f:X→Z is almost contraβθ-continuous.

(3) If f is weakly β-irresolute and g is almost contra βθ -continuous, then g ◦ f is almost contraβθ-continuous.

Theorem 2.12. If f:X→Y is a preβθ-closed surjection and g:Y→

Z is a function such that g ◦ f:X→Z is almost contraβθ-continuous,

then g is almost contraβθ-continuous.

Proof.Let V be any regular open set in Z Since g ◦ f is almost

contraβθ-continuous, f−1(g−1((V)))=(g ◦ f)−1(V)is βθ-closed

Sincefisapreβθ-closedsurjection, f(f−1(g−1((V))))=g−1(V)is

βθ-closed.Thereforegisalmostcontraβθ-continuous 

Theorem 2.13. Let X i:i∈}be any family of topological spaces If

f:X→X i is an almost contraβθ-continuous function, then Pr i ◦ f:

X→X i is almost contraβθ-continuous for each i∈, where Pr i is

the projection of

X i onto X i

Proof.Let U i be an arbitraryregular open set in X i Since Pr i is

continuousandopen,itisanR-mapandhenceP i−1(U i)isregular

openin

X i.Since fisalmost contraβθ-continuous, wehaveby

definition f−1(P−1i (U i))=(P i ◦ f)−1(U i)isβθ-closedinX

There-forePr i ◦ fisalmostcontraβθ-continuousforeachi∈ 

Definition 4. Afunctionf:X→Yiscalledweaklyβθ-continuous

ifforeach x∈X andeveryopen setV ofYcontainingf(x), there

existsaβθ-opensetUinXcontainingxsuchthatf(U)⊂ Cl(V)

Theorem 2.14. For a function f:X→Y, the following properties hold:

(1)If f is almost contra βθ-continuous, then it is weakly βθ

-continuous,

(2)If f is weaklyβθ-continuous and Y is extremally disconnected,

then f is almost contraβθ-continuous.

Proof.

(1)Let x ∈ X and V be any open set of Y containing f(x)

Since Cl(V) is a regular closed set containing f(x), by

Theorem2.2thereexistsaβθ-opensetUcontainingxsuch

thatf(U)⊂ Cl(V).Therefore,fisweaklyβθ-continuous

(2)LetV be aregular closedsubset ofY.SinceYis extremally

disconnected, we have that V is a regular open set of Y

and the weak βθ-continuity of f implies that f−1(V)⊂

βInt θ(f−1(Cl(V)))=βInt θ f−1(V) Therefore f−1(V) is βθ

-open in X This shows that f is almost contra βθ

-continuous 

Definition 5. Afunctionf:X→Yissaidtobe:

a)neatly (βθ, )-continuousifforeach x∈XandeachV ∈SO(Y,

f(x)), there is a βθ-open set U in X containing x such that

Int(f(U))⊂ Cl(V)

b) (βθ, )-openiff(U)∈SO(Y)foreveryβθ-opensetUofX

Theorem 2.15. If a function f: X → Y is neatly(βθ, )-continuous

and(βθ, )-open, then f is almost contraβθ-continuous.

Proof.Suppose that x ∈ X andV ∈ SO(Y, f(x)) Since f is neatly

(βθ, )-continuous, there exists a βθ-open setU of X containing

xsuch that Int(f(U))⊂ Cl(V) Byhypothesis, f is(βθ, )-open.This

impliesthat f(U)∈SO(Y).It followsthat f(U)⊂ Cl(Int(f(U)))⊂ Cl(V)

Thisshowsthatfisalmostcontraβθ-continuous 

3 Some fundamental properties

Definition 6[6,22]. Atopologicalspace(X,τ)issaidtobe:

(1)βθ-T0 (resp.βθ-T1) ifforanydistinctpairofpointsxandy

inX,thereisaβθ-opensetUinXcontainingxbutnotyor

(resp.and)aβθ-opensetVinXcontainingybutnotx

(2)βθ-T2 (resp β-T2 [7]) iffor every pairof distinct points x

andy,thereexisttwoβθ-open(resp.β-open)setsUandV

suchthatx∈U, y∈VandU∩V=∅

Theorem 3.1. For a topological space(X,τ),the following properties are equivalent:

(1) (X,τ)isβθ-T0;

(2) (X,τ)isβθ-T1;

(3) (X,τ)isβθ-T2;

(4) (X,τ)isβ-T2;

(5) For every pair of distinct points x, y ∈X, there exist U, V ∈

βO(X)such that x∈U, y∈V andβCl(U)∩βCl(V)=∅;

(6) For every pair of distinct points x, y ∈X, there exist U, V ∈

βR(X)such that x∈U, y∈V and U∩V=∅.

(7) For every pair of distinct points x, y∈X, there exist U∈βθO(X,

x)and V∈βθO(X, y)such thatβCl θ(U)∩βCl θ(V)=∅.

Proof. Itfollowsfrom([6],Remark3.2andTheorem3.4)

Recallthatatopologicalspace(X,τ)issaidtobe:

(i) WeaklyHausdorff[23](brieflyweakly-T2)ifeverypointofX

isanintersectionofregularclosedsetsofX (ii) -Urysohn [24] if for each pair of distinct points x and y

in X, there exist U ∈ SO(X, x) and V ∈ SO(X, x) such that

Cl(U)∩Cl(V)=∅ 

Theorem 3.2. If X is a topological space and for each pair of dis-tinct points x1 and x2 in X, there exists a map f of X into a Urysohn topological space Y such that f(x1)=f(x2)and f is almost contraβθ -continuous at x1and x2,then X isβθ-T2.

Proof. Letx1andx2 beanydistinctpointsinX.Thenby hypothe-sis,thereisaUrysohnspaceYandafunctionf:X→Y,which sat-isfiestheconditionsofthetheorem.Lety i=f(x i)fori=1,2.Then

y1=y2.SinceYisUrysohn,thereexistopensetsU y1 andU y2 ofy1

andy2,respectively,inYsuch thatCl(U y1)∩Cl(U y2)=∅.Sincefis almostcontraβθ -continuousatx i,thereexistsaβθ-opensetW x i

containingx iinXsuchthat f(W x i)⊂ Cl(U y i)fori=1,2.Hencewe getW x1∩W x2=∅sinceC l(U y1)∩C l(U y2)=∅.HenceXisβθ-T2 

Corollary 3.3. If f is an almost contra βθ-continuous injection of a topological space X into a Urysohn space Y, then X isβθ-T2.

Proof. Foreachpairofdistinct pointsx1 andx2 inX ,fisan al-mostcontra βθ-continuous function ofX into aUrysohn spaceY

suchthatf(x1)=f(x2)sincefisinjective.HencebyTheorem3.2,X

isβθ-T2 

Theorem 3.4.

(1) If f is an almost contraβθ-continuous injection of a topological space X into a s-Urysohn space Y, then X isβθ-T2.

(2) If f is an almost contraβθ-continuous injection of a topological space X into a weakly Hausdorff space Y, then X isβθ-T1.

Proof.

(1) LetYbe -Urysohn.Sincefisinjective,wehavef(x)=f(y)for anydistinct pointsxandyinX.SinceYis -Urysohn,there existV1 ∈SO(Y, f(x))andV2 ∈SO(Y, f(y))such thatCl(V1)∩

Cl(V2)=∅.Sincefisalmostcontraβθ-continuous,there ex-istβθ-opensetsU1andU2 inXcontainingxandy, respec-tively, such that f(U1)⊂ Cl(V1) and f(U2)⊂ Cl(V2) Therefore

U1∩U2=∅.ThisimpliesthatXisβθ-T2 (2) Since Yis weakly Hausdorff and fis injective,for any dis-tinct pointsx1 andx2 ofX,there existV1, V2 ∈RC(Y) such that f(x1) ∈ V1, f(x2)∈V1, f(x2) ∈ V2 and f(x1)∈V2 Since

fis almost contra βθ-continuous, by Theorem 2.2 f−1(V1)

and f−1(V2) are βθ-open sets and x1∈f−1(V1), x2∈/

f−1(V1), x2∈f−1(V2), x1∈/ f−1(V2) Then, there exists U1,

U2 ∈βθO(X)such thatx1∈U1⊂ f−1(V1), x2∈U1,x2∈U2⊂

f−1(V2)andx1∈U2.ThusXisβθ-T1 

showninthefollowingexample

Example 3.5. Let X={a , b , }, τ ={∅, X ,{a},{b},{a , b}} The

sub-sets a and b areβθ-closedin(X,τ)but a, b isnotβθ-closed

Recallthat atopologicalspaceiscalledaβθc-space[25]ifthe

unionofanytwoβθ-closedsetsisaβθ-closedset

Theorem 3.6. If f, g:X→Y are almost contraβθ-continuous

func-tions, X is aβθc-space and Y is s-Urysohn, then E={x∈X| f(x)=

g(x) }isβθ-closed in X.

Proof. Ifx∈X ࢨE,thenf(x) =g(x).SinceYis -Urysohn,there

ex-istV1∈SO(Y, f(x))andV2 ∈SO(Y, g(x))suchthatCl(V1)∩Cl(V2)=

∅ By the fact that f and g are almost contra βθ-continuous,

there exist βθ-open sets U1 and U2 in X containing x such that

f(U1)⊂ Cl(V1)andg(U2)⊂ Cl(V2).WeputU=U1∩U2.ThenUisβθ

-open in X Thus f(U)∩g(U)=∅ It follows that x∈βCl θ(E) This

showsthatEisβθ-closedinX

We say that the product space X=X1× × X n has Property

P βθ if A i is a βθ-open set in a topological space X i, for i=

1,2, n ,then A1× × A n is alsoβθ-open intheproduct space

X=X1× × X n 

Theorem 3.7. Let f1:X1→Y and f2:X2→Y be two functions, where

(1) X=X1× X2has the Property P βθ .

(2) Y is a Urysohn space.

(3) f1 and f2 are almost contra βθ-continuous Then { (x1, x2):

f1(x1)= f2(x2) } is βθ-closed in the product space X=X1×

X2

Proof. Let A denote the set { (x1, x2): f1(x1)= f2(x2) } In

or-der to show that A is βθ-closed, we show that (X1 × X2)\A

is βθ-open Let (x1, x2)∈A Then f1(x1) = f2(x2) Since Y is

Urysohn , there exist open sets V1 and V2 containing f1(x1)

andf2(x2), respectively,such thatCl(V1)∩Cl(V2)=∅.Since f i (i=

1,2)is almost contra βθ-continuous andCl(V i) is regular closed,

then f i−1(Cl(V i)) is a βθ-open set containing x i in X i (i=1,2).

Hence by (1), f1−1(Cl(V1))× f−1

2 (Cl(V2)) is βθ-open Furthermore

(x1, x2)∈f1−1(Cl(V1))× f−1

2 (Cl(V2))⊂(X1× X2) \A It follows that (X1× X2)\Aisβθ-open.ThusAisβθ -closedintheproductspace

X=X1× X2 

Corollary 3.8. Assume that the product space X × X has the Property

P βθ If f:X→Y is almost contraβθ-continuous and Y is a Urysohn

space Then A={ (x1, x2):f(x1)=f(x2)}isβθ -closed in the product

space X × X.

Theorem 3.9. Let f: X →Y be a function and g: X → X × Y the

graph function, given by g(x)=(x , f(x))for every x∈ X Then f is

almost contraβθ-continuous if g is almost contraβθ-continuous.

Proof. Letx ∈X andV be a regular open subset ofY containing

f(x) Then we have that X × V is regular open Since gis almost

contra βθ-continuous,g−1(X × V)= f−1(V) isβθ-closed.Hence f

isalmostcontraβθ-continuous 

Recall that for a function f: X → Y,the subset {(x, f(x)): x ∈

X ⊂ X × YiscalledthegraphoffandisdenotedbyG(f)

Definition 7. A function f: X → Y has a βθ-closed graph if for

each (x, y)∈(X × Y)\G(f), thereexistsU∈βθO(X, x)andanopen

setVofYcontainingysuchthat(U × V)∩G(f)=∅

Lemma 3.10. The graph, G(f)of a function f:X→Y isβθ-closed if

and only if for each(x, y) ∈(X × Y)\G(f)there exists U ∈βθO(X, x)

and an open set V of Y containing y such that f(U)∩V=∅.

Theorem 3.11. If f:X→Y is a function with aβθ-closed graph, then for each x∈X, f(x)=∩{Cl(f(U)):U∈βθO(X , x) }.

Proof. Supposethe theoremisfalse Then thereexists a y = f(x) such that y ∈ ∩ Cl(f(U)): U ∈βθO(X, x)} This implies that y ∈

Cl(f(U)),foreveryU∈βθO(X, x).SoV∩f(U)=∅,foreveryV∈O(Y,

y).whichcontradictsthehypothesisthatfisafunctionwithaβθ -closedgraph.Hencethetheorem 

Theorem 3.12. If f:X→Y is almost contraβθ-continuous and Y is Haudsorff, then G(f)isβθ-closed.

Proof. Let(x, y) ∈ (X × Y)\G(f) Then y = f(x) Since Y is Haus-dorff, there exist disjoint open sets V and W of Y such that

y ∈ V and f(x) ∈ W Then f(x)∈Y\Cl(W) Since Y\Cl(W) is a regular open set containing V, it follows that f(x)∈rker(V) and hencex∈/ f−1(rker(V)).Then by Theorem2.2(6)x∈/βCl θ(f−1(V) Thereforewehave(x , y)∈(X\ βCl θ((f−1(V)))× V⊂(X × Y) \G(f),

whichprovesthatG(f)isβθ-closed 

Theorem 3.13. Let f:X→Y have aβθ-closed graph.

(1) If f is injective, then X isβθ-T1.

(2) If f is surjective, then Y is T1.

Proof.

(1) Letx1 andx2 be anydistinctpointsinX.Then (x1,f(x2)) ∈ (X × Y)\G(f).Since fhasa βθ-closedgraph,thereexistU∈

βθO(X, x1)andanopensetVofYcontainingf(x2)suchthat

f(U)∩V=∅.ThenU∩f−1(V)=∅.Sincex2∈f−1(V), x2∈U ThereforeUisaβθ-opensetcontainingx1butnotx2,which provesthatXisβθ-T1

(2) Lety1 andy2 beanydistinct pointsinY.Since Yis surjec-tive, thereexists x ∈X such that f(x)=y1 Then (x, y2) ∈ (X × Y)\G(f).Since fhasa βθ-closedgraph,thereexistU∈

βθO(X, x) and an open set V of Y containingy2 such that

f(U)∩V=∅.Sincey1= f(x)andx∈U, y1∈f(U).Therefore

y1∈V,whichprovesthatYisT1 

Theorem 3.14. If f:X →Y has a βθ-closed graph and X is a βθ c-space, then f−1(K)isβθ-closed for every compact subset K of Y

Proof. LetKbeacompactsubsetofYandletx∈X\f−1(K).Then foreachy ∈K,(x, y)∈(X × Y)\G(f).SothereexistU y∈βθO(X, x) andanopensetV yofYcontainingysuchthat f(U y)∩V y=∅.The family V y:y∈K isanopencoverofKandhencethereisafinite subcover {V y i:i=1, , n} Let U=∩n

i=1U y i Then U ∈ βθO(X, x) andf(U)∩K=∅.HenceU∩f−1(K)=∅,whichprovesthat f−1(K)

isβθ-closedinX 

Definition 8. AtopologicalspaceXissaidtobe:

(1) stronglyβθC-compact[6]ifeveryβθ-closedcoverofXhas

afinitesubcover.(resp.A ⊂ XisstronglyβθC-compactifthe subspaceAisstronglyβθC-compact)

(2) nearly-compact[26] ifevery regular open coverof Xhas a finitesubcover

Theorem 3.15. If f: X→ Y is an almost contraβθ-continuous sur-jection and X is stronglyβθC-compact, then Y is nearly compact.

Proof. Let V α: α ∈ I be a regular open cover of Y Since f is almost contra βθ-continuous, we havethat {f−1(V α):α∈I} is a cover of X by βθ-closed sets Since X is strongly βθC-compact, there exists a finite subset I0 of Isuch that X={f−1(V α):α∈

I0}.SincefissurjectiveY={V α:α∈I0}andthereforeYisnearly compact

A topological space X is said to be almost-regular [27] if for eachregular closedset FofX andeachpoint x∈X\F, thereexist disjointopensetsUandVsuchthatF ⊂ Vandx∈U 

Theorem 3.16. If a function f:X→Y is almost contraβθ-continuous

and Y is almost-regular, then f is almostβθ-continuous.

Proof.Let x be an arbitrary point of X and V an open set of

Y containing f(x) Since Y is almost-regular, by Theorem 2.2 of

[27] there exists a regular open set W in Y containing f(x) such

that Cl(W)⊂ Int(Cl(V)) Since f is almost contra βθ -continuous,

andCl(W) is regular closed in Y, by Theorem 3.1 there exists U

∈ βθO(X, x) such that f(U)⊂ Cl(W) Then f(U)⊂ Cl(W)⊂ Int(Cl(V))

Hence,fisalmostβθ-continuous

The βθ-frontier ofa subset A,denoted by Fr βθ(A), is defined

as F βθ(A)=βCl θ(A) \ βInt θ(A), equivalently F βθ(A)=βCl θ(A)∩

βCl θ(X\A) 

Theorem 3.17. The set of points x∈X which f:(X,τ)→(Y,σ)is not

almost contra βθ-continuous is identical with the union of theβθ

-frontiers of the inverse images of regular closed sets of Y containing

f(x).

Proof.Necessity Suppose that f is not almost contra βθ

-continuousat a point x of X Then there exists a regular closed

setF ⊂ Y containingf(x) suchthat f(U) isnotasubset ofFfor

ev-eryU∈βθO(X, x) Hencewe haveU∩(X\f−1(F))=∅forevery

U∈βθO(X, x).It followsthat x∈βCl θ(X\f−1(F)).We alsohave

x∈f−1(F)⊂βCl θ(f−1(F)).Thismeansthatx∈F βθ(f−1(F))

Sufficiency Suppose that x∈F βθ(f−1(F)) for some F ∈ RC(Y,

f(x))Now,we assume that fis almost contra βθ-continuous at x

∈ X Then there exists U ∈ βθO(X, x) such that f(U)⊂ F

There-fore, we have x∈U ⊂ f−1(F) and hence x∈βInt θ(f−1(F))⊂ X\

F βθ(f−1(F)) Thisisacontradiction Thismeans thatfisnot

al-mostcontraβθ-continuous 

References

[1] C.W Baker , On contra almost β-continuous functions in topological spaces,

Kochi J Math 1 (2006) 1–8

[2] E Ekici , Almost contra-precontinuous functions, Bull Malaysian Math Sci Soc

27 (2006) 53–65

[3] T Noiri , V Popa , Some properties of almost contra-precontinuous functions, Bull Malaysian Math Sci Soc 28 (2005) 107–116

[4] M Caldas , S Jafari , Some properties of contra- β-continuous functions, Mem Fac Sci Kochi Univ Ser A (Math.) 22 (2001) 19–28

[5] S Jafari , T Noiri , On contra-precontinuous functions, Bull Malaysian Math Sci Soc 25 (2) (2002) 115–128

[6] M Caldas , On contra βθ-continuous functions, Proyecciones J Math 39 (4) (2013) 333–342

[7] A S Mashhour , M.E.A El-Monsef , S.N El-Deeb , On precontinuous and weak precontinuous mappings, Proc Math Phys Soc Egypt 53 (1982) 47–53

[8] N Levine , Semi-open sets and semi-continuity in topological spaces, Amer Math Monthly 68 (1961) 44–46

[9] M.E.A El-Monsef , S.N Ei-Deeb , R.A Mahmoud , β-open sets and β-continuous mappings, Bull Fac Assiut Univ 12 (1983) 77–90

[10] O Njastad , On some classes of nearly open sets, Pacific J Math 15 (1965) 961–970

[11] J.E Joseph , M.H Kwack , On s -closed spaces, Proc Amer Math Soc 80 (1980) 341–348

[12] T Noiri , Weak and strong forms of β-irresolute functions, Acta Math Hungar

99 (2003) 315–328

[13] E Ekici , Another form of contra-continuity, Kochi J Math 1 (2006) 21–29

[14] J Dontchev , Contra-continuous functions and strongly s-closed spaces, Inter- nat J Math Math Sci 19 (1996) 303–310

[15] J Dontchev , M Ganster , I Reilly , More on almost s-continuity, Indian J Math

41 (1999) 139–146

[16] D Andrijevic , Semi-preopen sets, Mat Vesnik 38 (1986) 24–32

[17] T Noiri , On almost continuous functions, Indian J Pure Appl Math 20 (1989) 571–576

[18] M.C Pal , P Bhattacharyya , Faint precontinuous functions, Soochow J Math 21 (1995) 273–289

[19] G.J Wang , On s -closed spaces, Acta Math Sinica 24 (1981) 55–63

[20] D Carnahan , Some properties related to compactness in topological spaces, University of Arkansas, 1973 Ph d thesis

[21] M Caldas , On θ- β-generalized closed sets and θ- β-generalized continuity in topolological spaces, J Adv Math Studies 4 (2011) 13–24

[22] M Caldas , Weakly sp- θ-closed functions and semipre-hausdorff spaces, Cre- ative Math Inform 20 (2) (2011) 112–123

[23] T Soundarajan , Weakly hausdorff space and the cardinality of topological spaces, general topology and its relation to modern analysis and algebra III, in: Proceedings Conference Kampur, 168, Academy Prague, 1971, pp 301–306

[24] S.P Arya , M.P Bhamini , Some generalizations of pairwise urysohn spaces, In- dian J Pure Appl Math 18 (1987) 1088–1093

[25] M Caldas , Functions with strongly β- θ-closed graphs, J Adv Stud Top 3 (2012) 1–6

[26] M.K Singal , A Mathur , On nearly compact spaces, Boll Un Mat Ital 4 (2) (1969) 702–710

[27] M.K Singal , S.P Arya , On almost-regular spaces, Glasnik Mat III 4 (24) (1969) 89–99