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SEMI-PRECONTINUOUS FUNCTIONS AND PROPERTIES OF GENERALIZED SEMI-PRECLOSED SETS
IN TOPOLOGICAL SPACES
G. B. NAVALAGI Received 12 June 2000
Andrijevi´c (1986) introduced the class of semi-preopen sets in topological spaces. Since then many authors including Andrijevi´c have studied this class of sets by defining their neighborhoods, separation axioms and functions. The purpose of this paper is to pro- vide the new characterizations of semi-preopen and semi-preclosed sets by defining the concepts of semi-precontinuous mappings, semi-preopen mappings, semi-preclosed map- pings, semi-preirresolute mappings, pre-semipreopen mappings, and pre-semi-preclosed mappings and study their characterizations in topological spaces. Recently, Dontchev (1995) has defined the concepts of generalized semi-preclosed (gsp-closed) sets and gener- alized semi-preopen (gsp-open) sets in topology. More recently, Cueva (2000) has defined the concepts like approximately irresolute, approximately semi-closed, contra-irresolute, contra-semiclosed, and perfectly contra-irresolute mappings using semi-generalized closed (sg-closed) sets and semi-generalized open (sg-open) sets due to Bhattacharyya and Lahiri (1987) in topology. In this paper for gsp-closed (resp., gsp-open) sets, we also intro- duce and study the concepts of approximately semi-preirresolute (ap-sp-irresolute) map- pings, approximately semi-preclosed (ap-semi-preclosed) mappings. Also, we introduce the notions like contra-semi-preirresolute, contra-semi-preclosed, and perfectly contra-semi- preirresolute mappings to study the characterizations of semi-pre-T1/2spaces defined by Dontchev (1995).
2000 Mathematics Subject Classification: 54A05, 54C08, 54C10, 54D10, 54G05.
1. Introduction. In the literature the notions of semi [14] (resp., pre [17],α[18])- continuous mappings, semi [5] (resp., pre [17],α[18])-open mappings, semi [21] (resp., pre [12],α[18])-closed mappings, irresolute [10] (resp., preirresolute [23],α-irresolute [16])–mappings were introduced and studied using semi [14] (resp., pre [17],α[20])- open sets of and semi [6,9] (resp., pre [12], α[16, 18])–closed subsets ofX. In this paper, we introduce the concepts of semi-precontinuous mappings, semi-preopen mappings, semi-preclosed mappings, semi-preirresolute mappings, pre-semipreopen mappings, and pre-semi-preclosed mappings and study their characterizations in topological spaces using the semi-preopen sets and semi-preclosed sets due to D. Andrijevic [3] (note thatβ-open sets in [1] are the same as the semi-preopen sets of D. Andrijevic [3]). Recently, in [11], Julian Dontchev has defined the concepts of gener- alized semi-preclosed (gsp-closed) sets and generalized semi-preopen (gsp-open) sets in topology. In this paper, using these sets, we also introduce and study the concepts of approximately semi-preirresolute mappings, and approximately semi-preclosed
mappings. Also, we introduce the notions of contra-semi-preirresolute, contra- semi-preclosed, and perfectly contra-semi-preirresolute mappings to study the char- acterizations of semi-pre-T1/2spaces defined by Julian Dontchev in [11].
2. Preliminaries. Throughout the present paper, the setsX, Y , Zalways mean topo- logical spaces andf:X→Y represents a single-valued function on which no separa- tion axioms are assumed unless explicitly stated. LetAbe a subset of a spaceX. The closure ofAand the interior ofAare denoted by clAand intA, respectively. The sub- setAofXis said to be (i) preopen [17] ifA⊂int clA, (ii) semi-open [14] ifA⊂cl intA, (iii) semi-preopen [3] ifA⊂cl int clA, and (iv)α-open [20] set ifA⊂int cl intA. The family of all preopen (resp., semi-open, semi-preopen, andα-open) sets ofX is de- noted by PO(X)(resp., SO(X), SPO(X), andα(X)). The family of all preopen (resp., semi-preopen) sets of X containing a pointx is denoted by PO(x)(resp., SPO(x)).
The complement of a semi-preopen (resp., preopen, semi-open, andα-open) set is called semi-preclosed (resp., preclosed, semi-closed, andα-closed) set ofX. The fam- ily of all semi-preclosed subsets ofXis denoted by SPF(X). A setMx⊂Xis said to be semi-preneighborhood [19] of a pointx inX if and only if there exists a semi- preopen setAcontainingxsuch thatA⊂Mx, the union of all semi-preopen sets that are contained inAis called the semi-preinterior [3] ofAand is denoted by(As)∗in [19] and the set∩{F⊂X|A⊂F andF is semi-preclosed set inX}is called the semi- preclosure [3] ofAand is denoted by(As)∗in [19]. A mappingf:X→Y said to be (i) semi-continuous [14] iff–1(A)∈SO(X)for every open setAinY, (ii) pre-continuous [17] iff−1(A)∈PO(X)for every open setAinY, (iii) semi-open [5] iff (A)∈SO(Y ) for every open setAinX, (iv) preopen [17] iff (A)∈PO(Y )for every open setAin X, (v) irresolute [10] iff−1(U )∈SO(X)for everyU∈SO(Y ), (vi) preirresolute [23] if f−1(U )∈PO(X)for everyU∈PO(Y ), (vii) pre-semiopen [10] iff (A)∈SO(Y )for every A∈SO(X), (viii) pre-semiclosed [22] iff (A)is a semi-closed set inY, for every semi- closed setAofX, (ix) semi-closed [6] iff (A)is semi-closed inY for each closed setA inXandXpreclosed [16] iff (A)is preclosed inY for every closed setAinX. More recently in [9], M. Caldas Cueva has defined the following notions: a mapf:X→Y is called (i) approximately irresolute (written as ap-irresolute) ifsclF ⊆f−1(O), when- ever Ois a semi-open subset ofY, F is sg-closed subset of X, andF ⊆f−1(O), (ii) approximately semi-closed (written as ap-semiclosed) map iff (B)⊆sintA, whenever Ais a sg-open subset ofY,Bis a semi-closed subset ofX, andf (B)⊆A, (iii) contra- irresolute iff−1(O)is semi-closed inXfor eachO∈SO(Y ), (iv) contra-presemiclosed iff (B)∈SO(Y )for each semi-closed setBofX, and (v) perfectly contra-irresolute if the inverse of every semi-open set inY is semi-clopen inX.
3. Semi-precontinuous functions. In this section, we introduce new weaker forms of continuity using semi-preopen sets and obtain their properties.
Definition3.1. A functionf:X→Y is called semi-precontinuous if the inverse image of each open set inY is a semi-preopen set inX.
Note3.2. Every continuous function is a semi-precontinuous function but not the converse; which is verified by the following example.
Example3.3. LetX= {a, b, c}, τ1= {∅,{a},{b},{a, b}, X}, Y = {1,2,3,4},τ2= {∅,{1},{1,2},{1,2,3}, Y}. A functionf:X→Yis defined byf (a)=1,f (b)=3,f (c)=2.
Here SPO(τ1)= {∅,{a},{b},{a, b},{a, c},{b, c}, X}. Then f is semi-precontinuous.
Butf is not continuous sincef−1({1,2})= {a, c}, which is not aτ1–open set.
Theorem3.4. Letf:X→Y be a single-valued function, whereXandY are topo- logical spaces. Then the following are equivalent:
(i) The functionfis semi-precontinuous.
(ii) For each pointp∈Xand each open setVinYwithf (p)∈V, there is aSPOset UinXsuch thatp∈U,f (U )⊆V.
(iii) The inverse of each closed set is semi-preclosed.
(iv) For each x ∈ X, the inverse of every neighborhood of f (x) is a semi- preneighborhood ofx.
(v) For each x ∈ X and each neighborhood Nx of f (x), there is a semi- preneighborhoodVofxsuch thatf (V )⊆Nx.
(vi) For each subsetAofX,f [(As)∗]⊆cl[f (A)].
(vii) For each subsetBofY,((f−1(B))s)∗⊆f−1(cl(B)).
Proof. (i)(ii). Letf (p)∈V andV⊂Y an open set, thenp∈f−1(V )∈SPO(τ);
sincef is semi-precontinuous. LetU=f−1(V ), thenp∈Uandf (U )⊂V.
Conversely, letVbe open inY andp∈f−1(V )thenf (p)∈V, there exists aUp∈ SPO(τ)such thatp∈Upandf (Up)⊂V. Thenp∈Up⊂f−1(V )andf−1(V )= ∪Up, but byNote 3.2,f−1(V )∈SPO(τ), which implies thatf is semi-precontinuous.
(i)(iii). Assumefis semi-precontinuous. LetBbe a closed subset ofY. ThenY−Bis open inYandf−1(Y−B)=X−f−1(B)∈SPO(τ), which implies thatf−1(B)is sp-closed.
Conversely, assume (iii). LetG be an open set inY thenY−Gis a closed set in Y. Thenf−1(Y−G)=X−f−1(G). Hencef−1(G)is SPO inXwhich implies thatf is semi–precontinuous.
(iii)⇒(iv). Assume (iii) forx∈X, letVbe the neighborhood off (x)thenf (x)∈W⊂ V, whereW=Y−F andF is closed inY. Consequently,f−1(W )is a semi-preopen set inX. Sincef−1(W )=f−1(Y−F )=X−f−1(F ),f−1(F )is sp-closed by hypothesis, and x∈f−1(W )⊂f−1(V ). Then by definition,f−1(V )is a semi-preneighborhood ofX.
(iv)⇒(v). Letx∈XandNxbe a neighborhood off (x). ThenV=f−1(Nx)is a sp-nhd ofxandf (V )=f (f−1(Nx))⊂Nx.
(v)⇒(ii). ForxinX, letWbe an open set containingf (x). ThenWis a neighborhood V ofxsuch thatx∈V andf (V )⊂W. Hence there exists a semi-preopen setAinX such thatx∈A⊂V, consequently,f (A)⊂f (V )⊂W.
(iii)(vi). Suppose that (iii) holds and letAbe a subset ofX. SinceA⊂f−1(f (A)), we haveA⊂f−1[cl(f (A))]. Now, clf (A)is a closed set inY and hencef−1{cl(f (A))}is a sp-closed set containingA, consequently,(As)∗⊂f−1{cl(f (A))}. Then,f [(As)∗]⊂ f f−1{cl(f (A))} ⊂cl[f (A)]. Conversely, suppose that (vi) holds for any subsetAof X. LetF be a closed subset ofY. Then,
f
f−1(F )
s
∗
⊂cl f
f−1(F )
⊂clF=F (3.1)
which implies that((f−1(F ))s)∗⊂f−1(F ). Consequently, the inverse of a closed set is semi-preclosed.
(vi)(vii). Obvious.
Recall that a subsetAofXis called regular open ifA=int clA, and that a mapping f :X→Y is called almost continuous [24] (written as a.c.S) if the inverse image of each regular open set is open.
Lemma3.5. Iff :X→Y is an open, a.c.S, and preirresolute map then f−1(B)∈ SPO(X)for eachB∈SPO(Y ).
Proof. SupposeBis an arbitrary semi-preopen set inY. Then there exists a pre- open setVinY such thatV⊂B⊂clV. Now, by the a.c.Sness off, we havef−1(V )⊂ f−1(B)⊂ f−1(clV ) =cl(f−1(V )) [2]. Since f is preirresolute and V is a preopen set inY,f−1(V )∈PO(X). Hence,f−1(V )⊂f−1(B)⊂(clf−1(V ))which implies that f−1(B)∈SPO(X).
Lemma3.6. Iff:X→Yis an open, a.c.S, and preirresolute mapping then the inverse image of each semi-preclosed set ofY is a semi-preclosed set inX.
The proof is similar toLemma 3.5.
Theorem3.7. Letf:X→Y be a mapping. Then the following are equivalent:
(i) The mappingf is semi-precontinuous.
(ii) For each subsetGofY,f−1(intG)⊂((f−1(G))s)∗.
Proof. (i)⇒(ii). Let G be any subset of Y. Then intG is an open set in Y and f−1(intG)is a semi-preopen set inX,fis semi-precontinuous. Asf−1(intG)⊂f−1(G), thenf−1(intG)⊂((f−1(G))s)∗.
The following lemma is proved in [3].
Lemma3.8. IfUis open andAis semi-preopen thenU∩Ais a semi-preopen set.
Lemma3.9. LetAbe a semi-preopen set in a spaceXand supposeA⊂B⊂clA, then Bis a semi-preopen set.
Proof. SinceAis a semi-preopen set inX, then there exists a preopen setUinX such thatU⊂A⊂clU. AsA⊂B,U⊂A⊂Bimplies thatU⊂B. Also, clA⊂cl(clU )=clU, and thus,B⊂clU. HenceU⊂B⊂clU, which implies thatBis a semi-preopen set.
Theorem3.10. IfA⊂X0⊂X andX0∈SPO(X). ThenA∈SPO(X)if and only if A∈SPO(X0).
Proof. Necessity: sinceA∈SPO(X), then there exists a preopen setU⊂Xsuch thatU⊂A⊂clU. Let clXand clX0denote, respectively, the closure operator inXand X0. Now,U⊂X0asX0⊂X. Then,U=U∩X0⊂A∩X0⊂X0∩clXU, orU⊂A⊂clX0U, since U=U∩X0 and U is preopen inX0 by [18, Lemma 2.3], then it follows that A∈SPO(X0).
The converse is easy and hence is omitted.
Lemma3.11(see [19]). For everyA∈SPO(X),clA=cl int clA.
Lemma3.12. Ais a semi-preopen set andA≠∅. Then,int clA≠∅.
Proof. LetAbe a semi-preopen set such thatA≠∅. Then byLemma 3.11, clA= cl int clA. If int clA= ∅then clA= ∅impliesA= ∅, which is in contradiction to the hypothesis. Hence, int clA≠∅.
Theorem3.13. Iff:X→Y is a semi-precontinuous map andX0is an open set inX, then the restrictionf /X0:X0→Y is semi-precontinuous.
Proof. Sincef is semi-precontinuous, for any open setV inY,f−1(V )is a semi- preopen set inX. Hence byLemma 3.8,f−1(V )∩X0is a semi-preopen set inXsince X0is an open set. Therefore, byTheorem 3.10,(f /X0)−1(V )=f−1(V )∩X0is a semi- preopen set inX0, which implies thatf /X0is semi-precontinuous.
We give the following definition.
Definition3.14. A coveru= {Uα|α∈∆}of subsets ofXis called a sp-cover if Uαis semi-preopen for eachα∈∆.
Now, we prove the following theorem.
Theorem 3.15. Let f : X →Y be a map and {Aα |α∈∆} a sp-cover ofX. If the restriction, f /Aα :Aα → Y is semi-continuous for each α∈ ∆, then f is semi- precontinuous.
Proof. SupposeV is an arbitrary open set inY. Then for each α∈∆, we have (f /Aα)−1(V )=f−1(V )∩Aα∈SPO(Aα)sincef /Aαis semi-precontinuous. Hence by Theorem 3.10, f−1(V )∩Aα∈SPO(X)for eachα∈∆. But, we know that arbitrary union of semi-preopen sets is a semi-preopen set, thus, we obtain that∪α∈∆[f−1(V )∩ Aα]=f−1(V )∈SPO(X). This implies thatfis a semi-precontinuous map.
Lemma3.16(see [13]). Let{Xα|α∈∆}be a family of topological spaces andΠAα
a subset of ΠXα, whereΠXαdenotes the product space. Then,
(i) intΠAα=ΠintAαifAα=Xαexcept for a finite number ofα∈∆andΠAα≠∅, (ii) clΠAα=ΠclAα.
Now, in view ofLemma 3.16, one can prove the following lemma.
Lemma3.17. LetX1andX2be topological spaces. IfAiis a semi-preopen set inXi
for eachi=1,2,thenA1×A2is a semi-preopen set in the product spaceX1×X2. The following theorem proved in [22] is the generalization ofLemma 3.17.
Theorem3.18(see [22]). Let{Xα|α∈∆}be a family of topological spaces,X= ΠXα, the product space, andA=Πnj=1Aαj×Πα≠αjXαa nonempty subset ofX, where n is a positive integer. Then,Aαj ∈SPO(Xαj) for eachj (1≤j≤n) if and only if A∈SPO(X).
Next, we prove the following theorem.
Theorem 3.19. LetXi and Yi be topological spaces andfi: Xi →Yi be a semi- precontinuous mapping fori=1,2. Then a mappingf:X1×X2→Y1×Y2defined by puttingf ((x1), (x2))is semi-precontinuous.
Proof. LetO1×O2⊂Y1×Y2, whereOiis open inYifori=1,2. Then,f−1(O1×O2)
=f−1(O1)×f−1(O2). Butf−1(O1), f−1(O2)are semi-preopen sets inX1and X2, re- spectively, thenf−1(O1)×f−1(O2)is semi-preopen inX1×X2byLemma 3.17. Now, ifO is any open set inY1×Y2, thenf−1(O)=f−1(∪Oα), where Oα is of the form Oα1×Oα2. Thenf−1(O)= ∪f−1(Oα), which is a semi-preopen set since arbitrary union of semi-preopen sets is a semi-preopen set. Hence by arguing as above,f−1(O1×O2) is a semi-preopen set inX1×X2. Hencef is semi-precontinuous.
The following theorem is the generalization ofTheorem 3.19, which can be proved in view of Theorems3.18and3.19.
Theorem3.20. Let{Xα|α∈∆}and{Yα|α∈∆}be any two families of topological spaces with the same index set∆. For eachα∈∆, letfα:Xα→Yαbe a mapping, then a mappingf:ΠXα→ΠYαdefined byf ((xα))=(fα(xα))is semi-precontinuous if and only iffαis semi-precontinuous for eachα∈∆.
4. Semi-preopen functions. We give the following definition.
Definition4.1. A functionf:(X, τ)→(Y , σ )is called semi-preopen if the image of each open set inXis a semi-preopen set inY.
Note that every open map is semi-preopen but not the converse, which is shown by the following example.
Example4.2. LetX= {p, q, r},τ= {∅,{p},{p, q}, X},Y = {a, b, c},σ = {∅,{a}, {b}{a, b}, Y}. Then it is clear thatτandσ are topologies onXandY, respectively. If f:(X, τ)→(Y , σ )is a map defined byf (p)=a,f (q)=c, andf (r )=b. It is clear that fis a semi-preopen mapping but it is not open mapping sincef ({p, q})= {a, c}∉σ. Also, note that every preopen (resp., semi-open) map is a semi-preopen map.
We recall that a mapping f : X → Y is called semi-preopen (in the sense of Cammaroto and Noiri [8]) iff (U )∈SPO(Y )for eachU∈SO(X). Clearly, every semi- preopen map (in the sense of Cammaroto and Noiri [8]) is a semi-preopen map as given byDefinition 4.1.
We recall the following lemma.
Lemma4.3(see [8]). The following are equivalent for a subsetAof a spaceX:
(i) A∈SPO(X).
(ii) A⊂cl(int(cl(A))).
(iii) A⊂sint(scl(A)).
Theorem4.4. A mappingf:X→Y is semi-preopen if and only if for every subset A⊂X,f (intA)⊂sint(scl(A)).
Proof. Letf be a semi-preopen map. We havef (intA)⊂f (A)for eachA⊂X and by hypothesisf (intA)is a semi-preopen set inY and byLemma 4.3,f (intA)⊂ sintscl(f (A)). Conversely, let the given condition holds true andGany open set in X. Thenf (G)=f (intG)⊂sintscl(f (G))which implies thatf (G)⊂sintscl(f (G)).
Thus,f (G)is a semi-preopen set inY byLemma 4.3and hencef is a semi-preopen map.
Theorem4.5. LetX, Y ,andZ be three topological spaces and letf :X→Y and g:Y →Zbe two mappings withg◦f:X→Zis a semi-preopen mapping. Then,
(i) Iffis continuous and surjective, thengis semi-preopen.
(ii) Ifgis preopen, preirresolute, and injective, thenf is semi-preopen.
Proof. (i) LetVbe an arbitrary open set inY. Sinceg◦fis semi-preopen andfis surjective theng(V )=g◦f{f−1(V )}is a semi-preopen set inZ. This shows thatgis a semi-preopen map.
(ii) Sincegis injective, we remark thatf (A)=g−1[g(f (A))]for every subsetAof X. LetUbe an arbitrary open set inX, then by hypothesis,g◦f (U )is a semi-preopen set inZ. Then byNote 3.2, we havef (U )=g−1(g◦f (U ))∈SPO(Y )which implies that f (U )is a semi-preopen set inY. Hence,fis a semi-preopen map.
5. Semi-preclosed functions. We recall the following definition.
Definition5.1. A functionf:X→Y is called semi-preclosed if the image of each closed set inXis a semi-preclosed set inY [22].
Note that every closed map is semi-preclosed but not the converse, which is shown by the following example.
Example 5.2. Let X = {a, b, c}, τ = {∅,{a},{a, b},{a, c}, X}, and σ = {∅,{a}, {a, b}, X}. Then f :(X, τ)→(X, σ ) is the identity mapping. Here, clearlyτ and σ are topologies onX. And, theτ-closed sets are{∅,{b},{c},{b, c}, X}, theσ-closed sets are{∅,{c},{b, c}, X}and SPF(σ )= {∅,{b},{c},{b, c}, X}.
Clearly,f is a semi-preclosed map but it is not closed sincef ({b})= {b}which is not aσ-closed set. Hence the example.
Note that every preclosed map (resp., semi-closed map) is a semi-preclosed map.
Thus, we state the following theorem.
Theorem5.3. A mapf:X→Y is semi-preclosed iff is both a semi-closed and a preclosed map.
Theorem5.4. Letf:(X, τ)→(Y , σ )be a mapping from a space(X, τ)into a space (Y , σ ). Thenfis semi-preclosed if and only if((f (A))s)∗⊂f ((As)∗), for each setAof (X, τ).
Proof. Letf be a semi-preclosed map andAany subset ofX. Thenf ((As)∗)∈ SPF(Y ). Asf (A)⊂f ((As)∗), it follows that(f (A))∗s ⊂f ((As)∗).
Conversely, assume thatF ∈SPF(X). Thenf (F )=f ((Fs)∗)⊃((f (F ))s)∗, thus we obtain that(f (F ))∗s =f (F ). Hence,f is a semi-preclosed map.
Next, we give the following definition.
Definition5.5. A mappingf :X→Y is said to bes-preclosed iff (F )∈SPF(Y ) for each semi-closed setF ofX.
Clearly, everys-preclosed map is semi-preclosed. But semi-closed, preclosed, and s-preclosed maps are, respectively, independent of each other.
Example 5.6. Let X = {p, q, r}, τ = {∅,{r},{q, r}, X}, and σ = {∅,{r},{p, r}, {q, r}, X}. Letf:(x, τ)→(X, σ )be a mapping defined byf (p)=q, f (q)=r, and f (r )=r. Then clearly,fiss-preclosed butfis not a preclosed map sincef ({p, q})= {q, r}which not preclosed in(X, σ ). This shows thats-preclosed and preclosed maps are independent of each other.
Recall the following lemma.
Lemma5.7(see [22]). Iff:X→Yis semi-preclosed, then for each subsetVofY and each open setUofXcontainingf−1(V ), there existsW∈SPO(Y )such thatV⊂Wand f−1(W )⊂U.
Now, we can prove it fors-preclosed maps in the following theorem.
Theorem 5.8. If f:X→Y is s-preclosed, then for each subsetV of Y and each semi-open setUofXcontainingf−1(V ), there existsW∈SPO(Y )such thatV⊂Wand f−1(W )⊂U.
6. Semi-preirresolute functions
Definition6.1. A functionf:X→Y is called semi-preirresolute if the inverse image of each semi-preopen set inY is a semi-preopen set inX.
Note that every semi-preirresolute map is semi-precontinuous but not the converse, which is shown by the following example.
Example 6.2. Let X = {a, b, c, d} and τ = {∅,{a},{b},{c},{a, b},{a, c},{b, c}, {b, d},{a, b, c},{a, b, d},{b, c, d}, X};Y= {m, n, l}, andσ= {∅,{m}, Y}. Letf:X→Y be a mapping defined byf (a)=m,f (b)=f (c)=l, andf (d)=n.
Then, clearlyf is semi-precontinuous but it is not a semi-preirresolute map since f−1({m, n})= {a, d}which is not a semi-preopen set in(Y , σ ).
Next, we characterize the semi-preirresolute mappings in the following theorem.
Theorem6.3. The following statements are equivalent for a functionf:X→Y: (i) fis semi-preirresolute.
(ii) For each pointxofXand each semi-preneighborhoodVoff (x), there exists a semi-preneighborhoodUofxsuch thatf (U )⊆V.
(iii) For eachx∈X and eachV ∈SPO(f (x)), there exists U∈SPO(x)such that f (U )⊆V.
Proof. (i)⇒(ii). Assumex∈XandV is a semi-preopen set inY containingf (x).
Sincef is a semi-preirresolute and letW=f−1(V )be a semi-preopen set inX con- tainingxand hencef (W )⊂f (f−1(V ))⊂V.
(ii)⇒(iii). Assume thatV ⊂Y is a semi-preopen set containingf (x). Then by (ii), there exists a semi-preopen setGsuch thatx∈G⊂f−1(V ). Therefore,x∈f−1(V )⊂ cl(f−1(V )). This shows that cl(f−1(V ))is a semi-preneighborhood ofx.
(iii)⇒(i). LetVbe a semi-preopen set inY, then cl(f−1(V ))is semi-preneighborhood of eachx∈f−1(V ). Thus, for eachxis a semi-preinterior point of cl(f−1(V ))which implies thatf−1(V )⊂int cl(f−1(V ))⊂cl int cl(f−1(V )). Therefore,f−1(V )is a semi- preopen set inXand hencefis a semi-preirresolute map.
We state the following theorems.
Theorem 6.4. If f :X→Y is a preopen and preirresolute mapping, thenf is a semi-preirresolute.
Theorem6.5. Iff:X→Yis semi-preirresolute andg:Y→Zis semi-precontinuous, theng◦fis a semi-precontinuous map.
Recall the following theorem.
Theorem 6.6(see [10, Theorem 1.1]). Iff :X→Y is continuous and open, then f−1(clA)=cl(f−1(A))for every subsetAofY.
Now, we prove the following theorem.
Theorem6.7. Letf:X→Y be a continuous open and preirresolute mapping, then f is a semi-preirresolute mapping.
Proof. LetA∈SPO(Y ), then there exists a preopen setU⊂Ysuch thatU⊂A⊂clU.
Then byTheorem 6.6,f−1(clU )=cl(f−1(U )). Also, we have f−1(U )⊂f−11(A)⊂f−1(clU )=cl
f−1(U )
. (6.1)
Sincefis a preirresolute map, thenf−1(U )is a preopen set inX, and hencef−1(A) is a semi-preopen set inX. Thus,f is a semi-preirresolute map.
One can easily prove the following theorem.
Theorem6.8. A mappingf:X→Y is semi-preirresolute if and only if for every semi-preclosed setF ofY,f−1(F )is a semi-preclosed set inX.
7. Semi-prehomeomorphisms
Definition 7.1. A bijective mappingf :(X, τ)→(Y , σ ) from a space X into a spaceY is called a semi-prehomeomorphism if bothfandf–1are semi-preirresolute mappings.
Now, we characterize the semi-prehomeomorphism in the following theorem.
Theorem7.2. Letf:(X, τ)→(Y , σ )be a bijective mapping from a spaceXinto a spaceY. Then the following are equivalent:
(i) fis a semi-prehomeomorphism.
(ii) f–1is a semi-prehomeomorphism.
Proof. (i)(ii). Sincef is a bijective map, bothf andf−1are semi-preirresolute functions.
Definition7.3. A property which is preserved under semi-prehomeomorphism is said to be a semi-pretopological property.
Clearly every homeomorphism is semi-prehomeomorphism.
8. Pre-semipreopen functions. In this section, we introduce the notion of pre- semipreopen mappings analogous to pre-semiopen mappings [10].
Definition8.1. A functionf :X→Y is called pre-semipreopen if the image of each semi-preopen set inXis a semi-preopen set inY.
Note that every pre-semipreopen map is semi-preopen but not the converse, which is shown by the following example.
Example 8.2. Let X = {a, b, c}, τ = {∅,{a},{a, b}, X}, Y = {p, q, r}, and σ = {∅,{p},{q},{p, q},{q, r}, Y}. Assume, a functionf:X→Y is defined byf (a)= {p}, f (b)= {q}, andf (c)= {r}. Then, clearlyf is a semi-preopen map but it is not pre- semipreopen sincef ({a, c})= {p, r}∉SPO(Y , σ ).
Remark8.3. (i) A pre-semipreopen map need not be open.
(ii) An open map need not be pre-semipreopen.
Theorem8.4. If a mappingf:X→Yis pre-semipreopen then((f (A))s)∗⊂f ((As)∗) for every subsetAofX.
Proof. Supposef is a pre-semipreopen map and Aany arbitrary subset of X.
Since(As)∗ is a semi-preopen set,f ((As)∗)is a semi-preopen set inY asf is a pre- semipreopen map. Hence, we obtain that((f (A))s)∗⊂f ((As)∗).
Theorem8.5. Letf :X→Y andg:Y →Z be two maps such thatg◦f is a pre- semipreopen map. Then,
(i) Iffis a semi-preirresolute surjection, thengis a pre-semipreopen map.
(ii) Ifgis a semi-preirresolute injection, thenfis a pre-semipreopen map.
Proof. (i) Let A be any semi-preopen set in Y. Since f is a semi-preirresolute map,f−1(A)is a semi-preopen set inX. Asg◦fis a pre-semipreopen map andf is surjective,g◦f (f−1(A))=g(A), which is a semi-preopen set inZ. This implies that gis a pre-semipreopen map.
(ii) As we claimed in (i), we can prove the second part easily.
9. Pre-semi-preclosed functions
Definition9.1. A functionf:X→Y is called pre-semi-preclosed if the image of each semi-preclosed set inXis a semi-preclosed set inY.
Note that every pre-semi-preclosed map is semi-preclosed but not the converse, which is shown by the following example.
Example9.2. Consider(X, τ),(Y , σ ), andfbe as defined inExample 8.2. By taking complements, one can show that f is a semi-preclosed map but not a pre- semi-preclosed map sincef ({b})= {q}∉SPF(Y , σ ).
Remark9.3. Both closed and pre-semi-preclosed maps are independent of each other.
One can prove the following theorem similar toLemma 5.7.
Theorem9.4. If a mappingf:X→Y is pre-semi-preclosed, then for each subsetB ofY and each semi-preopen setVinXcontainingf−1(B), there exists a semi-preopen setUinY containingBsuch thatf−1(U )⊂V.
Remark9.5. InTheorem 9.4, iff is also a surjective map then, the only if part holds true.
We prove the following theorem.
Theorem9.6. Letf :X→Y andg:Y →Z be two maps such thatg◦f is a pre- semi-preclosed map. Then,
(i) Iffis a semi-preirresolute surjection, thengis a pre-semi-preclosed map.
(ii) Ifgis a semi-preirresolute injection, thenfis a pre-semi-preclosed map.
Proof. We prove (ii) only. SupposeAis an arbitrary semi-preclosed set inX. Since g◦f is a pre-semi-preclosed map, theng◦f (A)is a semi-preclosed set in Z. Since g is a semi-preirresolute injective map, we haveg−1(g◦f (A))=f (A), which is a semi-preclosed set inY. This shows thatf is pre-semi-preclosed.
Recall that a mapf:X→Yis calledM-preclosed [23] if the image of each preclosed set is a preclosed set.
Finally, we prove the following theorem.
Theorem9.7. Iff:X→Y is a continuousM-preclosed injective map thenf is a pre-semi-preclosed map.
Proof. Letf be a continuousM-preclosed injective map andAa semi-preclosed set in X. Then, there exists a preclosed set F in X such that intF ⊂A⊂F and so f (intF )⊂f (A)⊂f (F ). Since f is a continuous injective map, int(f (F ))⊂f (intF ) and alsof is aM-preclosed map,f (F )is a preclosed set inY. Then, we obtain that int(f (F ))⊂f (A)⊂f (F )which implies thatf (A)is a semi-preclosed set inY. Thus, f is a pre-semi-preclosed map.
10. Generalized semi-preclosed sets and their mappings. We recall the following definition.
Definition10.1(see [15]). A subsetAof a spaceXis called a generalized closed set (written asg-closed) set if clA⊆UwheneverA⊆UandUis open.
Clearly, every closed set is ag-closed set. The complement of ag-closed set inXis called generalized open, that is,g-open [15] set. So, every open set is ag-open set.
Definition10.2(see [4]). A subsetAof a spaceX is called a semi-generalized closed set (written as sg-closed set) ifsclA⊆UwheneverA⊆UandUis semi-open.
Clearly every semi-closed set is a sg-closed set. The complement of a sg-closed set is called a semi-generalized open set, that is, a sg-open set [4]. Every semi-open set is a sg-open set.
Definition10.3(see [11]). A subsetAof a spaceXis called a generalized semi- preclosed (written as gsp-closed) set if(As)∗⊆UwheneverA⊆UandUis open.
Clearly, every semi-preclosed set is a gsp-closed set. The complement of a gsp- closed set is called generalized semi-preopen [11] (written as gsp-open). Every semi- preopen set is a gsp-open set. The family of all gsp-closed (resp., gsp-open) sets ofX is denoted by GSPF(X)(resp., GSPO(X)).
Next, we define the following mappings.
Definition 10.4. A function f : (X, τ)→ (Y , σ ) is called approximately semi- preirresolute (written as ap-sp-irresolute) if(Fs)∗ ⊆f−1(U ), whenever U is a semi- preopen subset of(Y , σ ),F is a gsp-closed subset of(X, τ), andF⊆f−1(U ).
Note that every semi-preirresolute (i.e., sp-irresolute) map is ap-sp-irresolute but not the converse, which is shown by the following theorem.
Theorem10.5. A functionf:(X, τ)→(Y , σ )is ap-sp-irresolute iff−1(U )is semi- preclosed in(X, τ)for everyU∈SO(Y , σ ).
Proof. LetF⊆f–1(U ), whereU∈SPO(Y , σ )andFis a gsp-closed subset of(X, τ).
Therefore, we have(Fs)∗⊆((f−1(U ))s)∗=f–1(U ). This implies thatf is an ap-sp- irresolute.
Theorem10.6. Letf :(X, τ)→(Y , σ ) be a map from a topological space(X, τ) into a topological space(Y , σ ). If the semi-preopen and semi-preclosed subsets of(X, τ) coincide, thenfis ap-sp-irresolute if and only iff−1(U )is semi-preclosed in(X, τ)for everyU∈SPO(Y , σ ).
Proof. Suppose thatf is an ap-sp-irresolute. LetAbe an arbitrary subset of X such thatA⊆U whereU∈SPO(X). Then by hypothesis(As)∗⊆(Us)∗=U. Thus, all subsets ofXare gsp-closed and hence all are gsp-open sets. Therefore, for anyOin SPO(Y ),f−1(O)is gsp-closed inX. Sincefis ap-sp-irresolute,((f−1(O))s)∗⊆f−1(O).
This shows that((f−1(O))s)∗=f−1(O),f−1(O)is a semi-preclosed set inX.
Converse follows fromTheorem 10.5.
Definition 10.7. A function f : (X, τ)→ (Y , σ ) is called approximately semi- preclosed (written as ap-sp-closed) iff (B)⊆(As)∗wheneverAis a gsp-open subset of(Y , σ ),Bis a semi-preclosed subset of(X, τ), andf (B)⊆A.
Note that every pre-semi-preclosed map is an ap-sp-closed map but not the con- verse, which is shown by the following theorem.
Theorem10.8. A functionf:(X, τ)→(Y , σ)is ap-semi-preclosed iff (B)∈SPO(Y , σ) for every semi-preclosed subsetBof(X, τ).
Proof. Letf (B)⊆A, whereBis a semi-preclosed subset of(X, τ)andAis gsp- open subset of (Y , σ ). Therefore, we have ((f (B))s)∗⊆(As)∗. Thenf (B)⊆(As)∗ which implies thatf is an ap-semi-preclosed map.
The easy proof of the following theorem is omitted.
Theorem10.9. Letf :(X, τ)→(Y , σ ) be a map from a topological space(X, τ) into a topological space(Y , σ ). If the semi-preopen and semi-preclosed subsets of(Y , σ ) coincide, thenf isap-semi-preclosed if and only iff (B)∈SPO(Y , σ ), for every semi- preclosed subsetBof(X, τ).
Also, we give the following definition.
Definition10.10. A functionf:(X, τ)→(Y , σ )is called contra-semi-preirresolute (written as contra-sp-irresolute) iff−1(O) is semi-preclosed in(X, τ)for eachO∈ SPO(Y , σ ).
Note that every semi-preirresolute (i.e., sp-irresolute) map is ap-sp-irresolute but not the converse, which is shown by the following definition.
Definition10.11. A functionf:(X, τ)→(Y , σ )is called contra-presemipreclosed (written as contra-presp-closed) iff (B)∈SPO(Y , σ ), for each semi-preclosed setBof (X, τ).
Note that every pre-semi-preclosed map is an ap-sp-closed map but not the con- verse, which is shown by the following theorem.
Theorem10.12. Letf:(X, τ)→(Y , σ )be a map. Then the following are equivalent:
(i) fis contra–sp-irresolute.
(ii) The inverse image of each semi-preclosed set inY is a semi-preopen set inX.
Next, we give the following definition.
Definition10.13. A subset of a spaceXis called semi-proclopen if it is both a semi-preopen and a semi-preclosed set.
Definition 10.14. A function f : (X, τ) → (Y , σ ) is called perfectly contra- semi-preirresolute (written as perfectly contra-sp-irresolute) if the inverse of every semi-preopen set inY is a semi-preclopen set inX.
We recall the following definition.
Definition10.15(see [11]). A spaceXis called semi-pre-T1/2if every gsp-closed set is semi-preclosed.
Next, we characterize the semi-pre-T1/2 spaces by using the ap-sp-irresolute and ap-semi-preclosed mappings.
Theorem10.16. Let(X, τ)be a topological space. Then the following statements hold:
(i) (X, τ)is a semi-pre-T1/2space.
(ii) For every space(Y , τ)and every mappingf:(X, τ)→(Y , σ),fis ap-sp-irresolute.
Proof. LetF be a gsp-closed subset of(X, τ)and suppose thatF⊆f−1(U )where U∈SPO(Y ). SinceXis semi-pre-T1/2space,Fis a semi-preclosed,F=(Fs)∗. Therefore, (Fs)∗⊆f−1(U ). This shows thatfis ap-sp-irresolute.
Theorem10.17. Let(Y , σ )be a topological space. Then the following statements hold:
(i) (Y , σ )is a semi-pre-T1/2space.
(ii) For every space (X, τ) and every mapping f : (X, τ) → (Y , σ ), f is ap- semi-preclosed.
Proof. The proof is similar toTheorem 10.16.
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G. B. Navalagi: Department of Mathematics, KLE Society’s, G. H. College, Haveri- 581110, Karnataka, India
E-mail address:[email protected]
