Pi-Set Connected Functions in Topological Spaces

Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 36, 1789 - 1798

π-Set Connected Functions in Topological Spaces

Ganes M. Pandya

School Of Petroleum Management Pandit Deendayal Petroleum University Gandhinagar-Gujrat, India ganes [email protected]

C. Janaki

Sree Narayana Guru College Coimbatore-Tamilnadu, India [email protected]

I. Arockiarani

Nirmala College for Women Coimbatore-Tamilnadu, India

Abstract

The aim of this paper is to introduce the concept of a new class of set connected functions called as π-set connected and study its properties.Furthermore we study and investigate relationship between π-set connected functions , seperation axioms and covering properties.

Mathematics Subject Classiﬁcation: 54C05, 54C10

Keywords: π-set connected, π-open sets, regular set-connected, clopen, πT1 space

1 Introduction and Preliminaries.

The notion of perfectly continuous function was introduced and studied by Noiri [10,11] and the notion of clopen functions was introduced by Reilly and Vamanamurthy [13]. Recently, a new class of functions called regular set- connected has been introduced by Dontchev, Ganster and Reilly [2].Ekici [4] 1790 G. M. Pandya, C. Janaki and I. Arockiarani extended the concept of regular set-connected functions to almost clopen functions. Throughout this paper, X and Y are topological spaces. Let A be a subset of X. We denote the interior and closure of a set by int(A) and cl(A) respectively. A subset A of a space X is said to be regular open if A = int( cl(A)). The family of all regular open (respectively regular closed, clopen) sets of X is denoted by RO(X) (respectively RC(X), CO(X)). A ﬁnite union of regular open sets is said to be π-open.The family of all π-open sets of X is denoted by πO(X).

Deﬁnition 1.1. A function f : ( X,τ) → (Y,σ) is said to be

1 . perfectly continuous [11] if f−1(V ) is clopen in X for every open set V of Y.

2 . set-connected [6] if f−1 (V ) is clopen in X for every V ∈ CO(Y ).

3 . regular set-connected[2] if f−1 (V ) is clopen in X for every V ∈ RO(Y ).

4 . almost π-continuous [3] if f −1(V) is π-closed in X for every regular closed set V of Y.

5 . almost clopen [4] if for each x ∈ X and each open set V in Y containing f(x), there exists a clopen set U in X containing x such that f(U) ⊂ int (cl(V)).

Deﬁnition 1.2. A space X is said to be

1. clopen T1 [4] if for each pair of distinct points x and y of X, there exist clopen sets U and V containing x and y respectively such that y ∈/ U and x ∈/ V.

2. clopen T2 [4]if for each pair of distinct points x and y in X, there exist disjoint clopen sets U and V in X such that x ∈ U and y ∈ V.

3. almost regular[15] if for each regular closed set F and each point x ∈/ F , there exist disjoint open sets U and V such that F ⊂ U and x ∈ V.

4. mildly normal [17]if for every pair of disjoint regular closed subsets F1 and F2 of X, there exist a disjoint open sets U and V such that F1⊂ U and F2 ⊂ V.

5. hyperconnected [19] if every non-empty open set is dense

6. mildly compact [18]if every clopen cover of X has a ﬁnite subcover. π-Set connected functions in topological spaces 1791

7. mildly Lindelof [18] if every cover of X by clopen sets has a countable subcover.

8. nearly compact[16] if every regular open cover of X has a ﬁnite subcover.

9. nearly countably compact [5]if every countable cover of X by regular open sets has a ﬁnite subcover.

10. nearly Lindelof if every cover of X by regular open sets has a countable subcover.

11. mildly countably compact[5] if every countable cover of X by clopen sets has a ﬁnite subcover.

2 π-set connected function.

Deﬁnition 2.1. A function f : X →Y is said to be π-set connected if f−1(V) ∈CO( X ) for every V ∈ πO( Y ). Theorem 2.2. Let (X, τ) and (Y,σ) be topological spaces. The following statement are equivalent for a function f : X →Y

1. f is π-set connected.

2. f−1(int(cl(G))) is clopen for every open subset G of Y.

3. f−1(cl(int(V))) is clopen for every closed subset V of Y.

4. f−1(F) is clopen for every F ∈ πC(Y).

Proof. (1) ⇒(2) Let G be any open subset of Y . Since int(cl(G)) is regular −1 open and hence π-open, by (1) it follows f (int(cl(G))) is clopen. (2)⇒(1) Let V be π-open in Y. By (2), f−1(int(cl(V)) is clopen in X and hence f is π-set connected.

(2) ⇒(3) Let V be a closed subset of Y. Then Y V is open. By (2),

−1 −1 −1

f (int(cl(Y V))) is clopen. Hence, f (int(cl(Y V))) = f ( int(Y int(V)

−1 −1 −1 )) = f (Y cl(int(V))) = X f (cl(int(V))). Hence f (cl(int(V))) is clopen. (3)⇒ (2) is obtained similarly . (1)⇔ (4) is Obvious. Theorem 2.3. If f : X →Yisπ-set connected function and A is any subset of X, then the restriction f/A:A→Yisπ-set connected functiion . Proof. Let V be a π-open set . By hypothesis f−1(V) is clopen in X. . We have f−1(V) ∩ A = (f /A)−1(V) is clopen in A. Hence f/A is π-set connected function. 1792 G. M. Pandya, C. Janaki and I. Arockiarani

Theorem 2.4. Let f : X →Y and g : Y →Z be functions. Then the following properties hold

1. If f and g are π-set connected functions, then g◦f:X→Zisπ-set connected function.

2. If f is π-set connected function and g is perfectly continuous, then g◦f:X → Z is perfectly continuous.

3. If f is set-connected and g is π-set connected then g◦fisπ-set connected.

4. If f is almost π-continuous and g is π-set connected function , then g◦f:X → Z is almost π-continuous.

Proof. (1) Let V be π-open in Z. By hypothesis ,g−1(V) is clopen in Y. Since fisπ-set connected function , f−1(g−1(V)) = ( g◦f)−1(V) is clopen. Therefore g◦fisπ-set connected function.. (2)Let V be open in Z. By hypothesis, g−1(V) is clopen in Y. Since f is π- set connected function, f−1(g−1(V)) is clopen in X .Therefore g ◦ f is perfectly continuous. (3)Let V be π-open in Z. Since g is π-set connected, g−1(V) is clopen in Y. Since f is set connected, f−1(g−1(V)) is clopen in X. hence g◦fisπ-set connected. (4)Let V be regular open in Z. Since g is π-set connected, g−1(V) is clopen in Y. f−1(g−1(V)) is π-open in Z. Then g◦f is almost-π-continuous.

Theorem 2.5. If f : X →Y is a surjective open and closed function and g : Y →Z is a function such that g◦f:X→Zisπ- set connected, then g is π-set connected.

Proof. Let V be π-open in Z. (g◦f)−1(V) is clopen in X. f−1(g−1(V)) is clopen in X. Since f is surjective open and closed, f (f−1(g−1(V) )) = g−1(V )is clopen.Therefore g is π-set connected.

3 Separation Axioms and Graphs

Deﬁnition 3.1. A Space X is said to be πT1 if for each pair of distinct points x and y of X , there exist π-open sets U and V containing x and y respectively such that y∈/ U and x ∈/ V.

Theorem 3.2. If f : X →Yisaπ-set connected injection and Y is πT1, then X is clopen T1 π-Set connected functions in topological spaces 1793

Proof. Since Y is πT1 for any distinct points x and y in X, there exist V,W ∈ πO(Y) such that f(x) ∈V , f(y) ∈/V , f(x) ∈/W, f(y) ∈ W. Since f is π-set connected, f−1(V) and f−1(W ) are clopen in X. Furthermore y ∈/f−1(V ) and x −1 ∈/f (W). This shows that X is clopen T1.

Theorem 3.3. If f : X →Yisaπ-set connected injection and Y is T2, then X is clopen T2.

Proof. For any pair of distinct points x and y in X, there exist disjoint open sets U and V in Y such that f(x) ∈U and f(y) ∈ V. Since f is π-set connected ,f−1(int(cl(U) )) and f−1(int(cl(V))) is clopen in X containing x and y respec- ctively. We have f−1(int(cl(U )) ∩ f−1(int(cl(V)) = φ . This shows that X is clopen T2 .

Theorem 3.4. If f : X→Y and g : X→Ybeπ-set connected functions and Y is Hausdorﬀ , then E = {x ∈ X : f(x)=g(x)} is closed in X. ∈ Proof. If x X E then it follows that f(x) = g(x) .Since Y is Hausdorﬀ, there exist open sets V and W such that f(x) ∈V , g(x) ∈ W and V ∩ W = φ. Since f and g are π-set connected, f−1(int(cl(V))) and g−1(int(cl(W))) are clopen in X with x∈ f−1(int(cl(V))) and x ∈ g−1(int(cl(W))). Set O = f−1(int(cl(V)))∩ g−1(int(cl(W))). Then O is open. Therefore f(O) ∩ g(O) = φ and it follows that x ∈/ cl(E). This shows that E is closed in X.

Theorem 3.5. If f is a π-set connected injective open function from a regular space X onto a space Y , then Y is almost regular.

Proof. Let F be regular closed set in Y and let y ∈/F. Take y = f(x). Since f is π -set connected function , f−1(F) is a clopen set.Take G = f−1(F) we have x∈/ G . Since X is regular, there exist disjoint open sets U and V such that G ⊂ U and x ∈ V. We obtain f(G) ⊂ f(U) and y = f(x) ∈ f(V) such that f(U) and f(V) are disjoint open sets.

Theorem 3.6. If f is a π-set connected injected open function from a normal space X onto a space Y, then Y is mildly normal.

Proof. Let F1and F2be disjoint regular closed subsets of Y . Since f is π-set −1 −1 −1 connected function, f (F1 ) and f (F2) are clopen sets. Take U = f (F1) −1 and V = f (F2). We have U ∩ V=φ . Since X is normal, there exist disjoint open sets A and B such that U ⊂ A and V ⊂ B. We obtain F1= f (U) ⊂ f (A) and F2 = f(V) ⊂ f(B) such that f(A) and f(B) are disjoint open sets . Thus Y is mildly normal.

Theorem 3.7. If X is connected and f : X → Yisπ -set connected and surjective , then Y is hyperconnected. 1794 G. M. Pandya, C. Janaki and I. Arockiarani

Proof. Assume that Y is not hyperconnected. Then there exist an open set V such that V is not dense in Y. Then there exist disjoint non-empty regular

open subsets B1and B2 in Y, namely int(cl(V)) and Y cl(V). Since f is π- −1 −1 set connected and onto A1=f (B1) and A2=f (B2) are disjoint non-empty clopen subsets of X. X is connected implies that A1and A2must intersect. Hence Y is hyperconnected. Theorem 3.8. Let f : X → Ybeaπ-set connected surjective function. Then the following statements hold:

1. If X is mildly compact, then Y is nearly compact.

2. If X is mildly Lindelof, then Y is nearly Lindelof.

3. If X is mildly countably compact, then Y is nearly countably compact.

Proof. Straight forward. Deﬁnition 3.9. A space X is said to be 1. π-closed if every π-closed cover of X has a ﬁnite subcover.

2. π-Lindelof if every cover of X by π-closed sets has a countable subcover.

3. countably π-closed if every countable cover of X by π-closed sets has a ﬁnite subcover.

Theorem 3.10. Let f : X → Ybeaπ-set connected surjective function .Then the following statements hold.

1. If X is mildly compact, then Y is π-closed.

2. If X is mildly Lindelof, then Y is π-connected.

3. If X is mildly countably compact, then Y is countably π-connected.

Proof. Let {Vα : α ∈ I}be any π -closed cover of Y. Since f is π -set connected, −1 then {f (Vα):α ∈ I} is clopen cover of X and hence there exist a ﬁnite subset −1 Ioof I such that X = ∪{f (Vα):α ∈ Io}. Therefore Y = ∪{Vα : α ∈ Io}and Yisπ -closed. Proof of (2) and (3) being entirely analogous Corollary 3.11. Let f : X → Ybeπ-set connected surjection function. Then the following statements hold.

1. If X is compact ( or π-closed ) then Y is π -closed. π-Set connected functions in topological spaces 1795

2. If X is Lindelof( π-Lindelof) then Y is π -Lindeloﬀ.

3. If X is countably compact ( countably-π-closed) then Y is countably π -closed.

Theorem 3.12. If f : X → Yisaπ-set connected function and Y is Haus- dorff, then E = {(x, y) ∈ X × X : f(x)=f(y)} is closed in X× X. Proof. Let (x, y) ∈ (X× X)\ E. It follows that f(x) =f( y). Since Y is Hausdorff there exist open sets V and W containing f(x) and f(y), respectively,such that V ∩ W=φ .Since f is π set-connected, f−1(int(cl(W))) are clopen in X with x∈ f−1(int(cl(V))) and y∈ f−1(int(cl(W))) . Take U = f−1(int(cl(V))) and G = f−1 (int(cl(W))) . Hence (U × G) ∩ E=φ.U× GisopeninX×X containing (x,y). This means that E is closed in X× X. Definition 3.13. A graph G(f) of a function f : X →Y is said to be co-π- closed if for each (x, y) ∈(X × Y)\ G(f), there exist a clopen set U of X and aV∈ πO(Y ) such that (x, y) ∈ U × V and (U × V)\ G(f) = φ. Theorem 3.14. If f : X → Yisπ set-connected and Y is Hausdorff, then G(f) is co-π-closed in X × Y. Proof. Let (x; y) ∈ (X × Y)\ G(f), then f(x) = y. Since Y is Hausdorff, there exist open sets U and V such that f(x) ∈ U, y ∈ V and U ∩ V=φ. Since f is π- set connected, f−1(int(cl(U))) is a clopen set in X such that x ∈ f−1(int(cl(U))). Take A = f−1(int(cl(U))). We have f(A) ⊂ int(cl(U)). Therefore, we obtain y∈ int(cl(V )) ∈ RO(Y ) and hence in πO(Y) and f(A) ∩ int(cl(V )) = φ. This shows that G(f) is co-π-closed. Theorem 3.15. Let f : X × Y has a co-π-closed graph G(f). If f is injective, then X is clopen T1. Proof. Let x and y be any two distinct points of X. Then, we have (x, f(y)) ∈ (X ×Y)\G(f). By co-π-closedness of graph G(f), there exist a clopen set U of X and V ∈ πO(Y ) such that (x, f(y)) ∈ U × V and (U × V)∩ G(f) = φ. Then, we have f(U) ∩ V=φ, hence U ∩ f−1(V ) = φ. Therefore, we have y ∈/ U. This implies that X is clopen T1. Theorem 3.16. Let f : X → Y has a co-π-closed graph G(f). If f is a surjective open function, then Y is T2

Proof. Let y1 and y2 be any distinct points of Y . Since f is surjective, f(x) = y1 for some x ∈ X and (x, y) ∈ (X × Y)\G(f). By co-π-closedness of graph G(f), there exist a clopen set U of X and V ∈ πO(Y ) such that (x. y2) ∈ U × V and (U × V)∩ G(f) = φ. Then, we have f(U)∩ V=φ. Since f is open, then f(U) is open such that f(x) = y1∈ f(U). This implies that Y is T2. 1796 G. M. Pandya, C. Janaki and I. Arockiarani

Theorem 3.17. Let ( X,τ ) and ( Y,σ ) be topological spaces. The following

statements are equivalent for a function f : X Y

1. f is almost clopen;

2. for each x ∈ X and each π-open set G in Y containing f(x), there exists a clopen set U in X containing x such that f(U)⊂ G;

3. for each x ∈ X and each regular closed set F in Y not containing f(x), there exists a clopen set U in X not containing x such that f−1(F) ⊂ U;

4. the inverse image of every π-open set of Y is a union of clopen sets of X ;

5. the inverse image of every π-closed set of Y is an intersection of clopen sets of X.

Proof. (1)⇒ (2).Let G be any π-open set in Y containing f(x).Since G is open, then by (1), it follows that there exists a clopen set U in X containing x such that f(U) ⊂ int (cl(G)) = G (2)⇒ (1)Let G be any open set in Y containing f(x). Since int(cl(G)) is π -open, then by (2), it follows that there exists a clopen set U in X containing x such that f(U) ⊂ int (cl(G)). (2)⇒ (3) Let F be any π-closed set in Y not containing f(x).Then, Y\Fisa π -open set containing f(x). By (2), there exists a clopen set U in X containing x such that f(U) ⊂ Y\F. Hence, U ⊂ f−1(Y \ F) ⊂ X\ f−1(F) and then f−1(F) ⊂ X \U. Take H = X \ U. We obtain that H is a clopen set in X not containing x. (3)⇒ (2). Straight Forward. (2)⇒ (4). Let G be any π-open set in Y and let x ∈ f−1(G). Since f(x) ∈ G, −1 by (2), there exists a clopen set Ux in X containing x such that Ux ⊂ f (G). −1 −1 Hence, we have f (G) = ∪ Ux where x∈f (G). The converse can be shown easily.

(4)⇐⇒ (5). This can be obtained similarly. Theorem 3.18. Let f : X Y and g : Y Z be functions. Then the

following properties hold : 1. If f is π-set connected and g is almost clopen, then gîf:X Z is almost

clopen. 2. If f is almost clopen and g is π-set connected, then gîf:X Z is almost clopen. π-Set connected functions in topological spaces 1797

Proof. (1). Let V be any open set in Z containing (g î f)(x). Since g is almost clopen, there exists a clopen set U in Y containing f(x) such that g(U) ⊂ int(cl(V)).Since f is π-set connected f−1(U) is clopen in X. Take G = f−1(U).

⊂ ⊂ î Then f(G) U. This shows that (gîf)(G) int cl (V ) .Therefore g fis almost clopen. (2)The proof is similar.

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Received: January, 2010