© Hindawi Publishing Corp.
A NOTE ON
θ-GENERALIZED CLOSED SETS
C. W. BAKER
(Received 22 September 1999)
Abstract.The purpose of this note is to strengthen several results in the literature concerning the preservation ofθ-generalized closed sets. Also conditions are established under which images and inverse images of arbitrary sets are θ-generalized closed. In this process several newweak forms of continuous functions and closed functions are developed.
2000 Mathematics Subject Classification. Primary 54C10.
1. Introduction. Recently Dontchev and Maki [5] have introduced the concept of a θ-generalized closed set. This class of sets has been investigated also by Arockiarani et al. [1]. The purpose of this note is to strengthen slightly some of the results in [5] concerning the preservation ofθ-generalized closed sets. This is done by using the notion of aθ-c-closed set developed by Baker [2]. These sets turn out to be a very natural tool to use in investigating the preservation ofθ-generalized closed sets. In this process we introduce a new weak form of a continuous function and a new weak form of a closed function, calledθ-g-c-continuous andθ-g-c-closed, respectively. It is shown that θ-g-c-continuity is strictly weaker than strongθ-continuity and that θ-g-c-closed is strictly weaker thanθ-g-closed.
2. Preliminaries. The symbolsXandY denote topological spaces with no separa-tion axioms assumed unless explicitly stated. IfAis a subset of a spaceX, then the closure and interior ofAare denoted by Cl(A)and Int(A), respectively. Theθ-closure ofA[8], denoted by Clθ(A), is the set of allx∈Xfor which every closed neighborhood ofxintersectsAnontrivially. A setAis calledθ-closed ifA=Clθ(A). Theθ-interior of A[8], denoted by Intθ(A), is the set of allx∈Xfor whichAcontains a closed neigh-borhood ofx. A setAis said to beθ-open provided thatA=Intθ(A). Furthermore, the complement of aθ-open set isθ-closed and the complement of aθ-closed set is θ-open.
Definition2.1(Dontchev and Maki [5]). A setAis said to beθ-generalized closed (or brieflyθ-g-closed) provided that Clθ(A)⊆UwheneverA⊆UandUis open. A set is calledθ-generalized open (or brieflyθ-g-open) if its complement isθ-generalized closed.
Theorem2.2(Dontchev and Maki [5]). A setAisθ-g-open if and only ifF⊆Intθ(A) wheneverF⊆AandFis closed.
Definition2.3(Dontchev and Maki [5]). A functionf:X→Y is said to beθ-g -closed provided thatf (A)isθ-g-closed inY for every closed subsetF ofX.
Definition2.4(Dontchev and Maki [5]). A functionf:X→Y is said to beθ-g -irresolute (θ-g-continuous), if for everyθ-g-closed (closed) subsetAofY,f−1(A)is θ-g-closed inX.
Definition2.5(Noiri [7]). A functionf:X→Yis said to be stronglyθ-continuous provided that, for everyx∈Xand every open neighborhoodV off (x), there exists an open neighborhoodUofxfor whichf (Cl(U))⊆V.
3. Sufficient conditions for images ofθ-g-closed sets to beθ-g-closed. Dontchev and Maki [5] proved that theθ-g-closed, continuous image of aθ-g-closed set isθ-g -closed. In this section, we strengthen this result by replacing both theθ-g-closed and continuous requirements with weaker conditions. Our replacement for theθ-g-closed condition uses the concept of aθ-c-open set from [2].
Definition3.1(Baker [2]). A setAis said to beθ-c-closed provided there is a set Bfor whichA=Clθ(B).
We define a functionf:X→Yto beθ-g-c-closed iff (A)isθ-g-closed inYfor every θ-c-closed setAinX. Sinceθ-c-closed sets are obviously closed,θ-g-closed implies θ-g-c-closed. The following example shows that the converse implication does not hold.
Example3.2. LetX= {a,b,c}have the topologyτ= {X,∅,{a},{a,b},{a,c}}and letf:X→X be the identity mapping. Since theθ-closure of every nonempty set is X,f is obviouslyθ-g-c-closed. However, sincef ({c})fails to beθ-g-closed,f is not θ-g-closed.
Theorem3.3. Iff:X→Y is continuous andθ-g-c-closed, thenf (A)isθ-g-closed inY for everyθ-g-closed setAinX.
Proof. AssumeAis aθ-g-closed subset ofXand thatf (A)⊆V, whereV is an open subset ofY. ThenA⊆f−1(V), which is open. SinceAisθ-g-closed, Clθ(A)⊆ f−1(V)and hencef (Clθ(A))⊆V. Because Clθ(A)isθ-c-closed andf isθ-g-c-closed, f (Clθ(A)) is θ-g-closed. Therefore Clθ(f (Clθ(A))) ⊆ V and hence Clθ(f (A)) ⊆
Clθ(f (Clθ(A)))⊆V, which proves thatf (A)isθ-g-closed.
Corollary3.4(Dontchev and Maki [5]). Iff:X→Y is continuous andθ-g-closed, thenf (A)isθ-g-closed inY for everyθ-g-closed subsetAofX.
Definition3.5. A functionf :X→Y is said to be approximatelyθ-continuous (or brieflya-θ-continuous) provided that Clθ(A)⊆f−1(V)wheneverA⊆f−1(V ),Ais θ-g-closed, andV is open.
The proof ofTheorem 3.3is easily modified to obtain the following result.
Theorem3.6. Iff :X→Y is a-θ-continuous andθ-g-c-closed, thenf (A)isθ-g -closed inY for everyθ-g-closed setAinX.
Obviously continuity impliesa-θ-continuity and the following example shows that a-θ-continuity is strictly weaker than continuity.
Example3.7. Let(X,τ)be the space inExample 3.2and letσ= {X,∅,{b}}. Then the identity mappingf:(X,τ)→(X,σ )is not continuous but isa-θ-continuous.
In [4] Dontchev defined a function to be contra-continuous provided that inverse images of open sets are closed. We modify this concept slightly to obtain aθ -contra-continuous function.
Definition3.8. A functionf:X→Y is said to beθ-contra-continuous if for every open subsetVofY,f−1(V)isθ-closed.
If the continuity requirement inTheorem 3.3is replaced withθ-contra-continuity, then a much stronger result is obtained. The step in the proof ofTheorem 3.3where we obtain Clθ(A)⊆f−1(V)nowholds for every setA, becausef−1(V )is θ-closed. Therefore we have the following theorem.
Theorem3.9. Iff :X→Y isθ-contra-continuous andθ-g-c-closed, thenf (A)is θ-g-closed inY for every subsetAofX.
4. Sufficient conditions forθ-g-irresoluteness. Dontchev and Maki [5] proved that a stronglyθ-continuous, closed function isθ-g-irresolute. We strengthen this result slightly by replacing strongθ-continuity and closure with weaker conditions.
We define a functionf:X→Y to beθ-g-c-continuous provided that, for everyθ-c -closed subsetAofY,f−1(A)isθ-g-closed. Since strongθ-continuity is equivalent to the requirement that inverse images of closed sets beθ-closed [6], strongθ-continuity obviously impliesθ-g-c-continuity. The function inExample 3.2isθ-g-c-continuous but not stronglyθ-continuous.
Theorem4.1. Iff:X→Y isθ-g-c-continuous and closed, thenfisθ-g-irresolute. Proof. AssumeA⊆Y isθ-g-closed and thatf−1(A)⊆U, whereUis open. Then X−U⊆X−f−1(A)and we see thatf (X−U)⊆Y−A. SinceAis θ-g-closed,Y−A is θ-g-open. Also, sincef is closed, f (X−U)is closed. Thus f (X−U)⊆Intθ(Y−
Corollary4.2(Dontchev and Maki [5]). Iff:X→Y is stronglyθ-continuous and closed, thenf isθ-g-irresolute.
Obviouslyθ-g-continuity impliesθ-g-c-continuity. Therefore we have the following result.
Corollary4.3. Iff:X→Y isθ-g-continuous and closed, thenfisθ-g-irresolute.
The function inExample 3.2isθ-g-c-continuous but notθ-g-continuous.
Theorem 4.1 can be strengthened in much the same way as Theorem 3.3 was strengthened by replacing the closure requirement with a weaker condition.
Definition 4.4. A function f : X→Y is said to be approximatelyθ-closed (or brieflya-θ-closed) provided thatf (F)⊆Intθ(A)wheneverf (F)⊆A,F is closed, and Aisθ-g-open.
Note that, under ana-θ-closed function, images of closed sets interact withθ-g -open sets in the same manner as closed sets. Obviously closed functions are a-θ -closed. The inverse of the function inExample 3.7isa-θ-closed but not closed. The proof of the following theorem is analogous to that ofTheorem 4.1.
Theorem4.5. Iff:X→Y isθ-g-c-continuous anda-θ-closed, thenfisθ-g -irreso-lute.
Finally,Theorem 4.1can be modified by replacing the requirement that the func-tion be closed with a variafunc-tion of a contra-closed funcfunc-tion. Contra-closed funcfunc-tions, introduced by Baker [3], are characterized by having open images of closed sets.
Definition4.6. A functionf:X→Y is said to beθ-contra-closed provided that f (F)isθ-open for every closed subsetFofX.
Theorem4.7. Iff:X→Y isθ-g-c-continuous andθ-contra-closed, then for every subsetAofY f−1(A)isθ-g-closed (and hence alsoθ-g-open).
The proof ofTheorem 4.7follows that ofTheorem 4.1, except that the stepf (X−
U)⊆Intθ(Y−A)holds for every subsetAofY becausef (X−U)isθ-open. References
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C. W. Baker: Department of Mathematics, Indiana University Southeast, New Albany, IN47150, USA
