ON WEAK FORMS OF SEMI-OPEN AND SEMI-CLOSED FUNCTIONS

Miguel Caldas and Govindappa Navalagi

Abstract. Weakly semi-open functions between topological spaces are defined as dual to the weak semi-continuous. Connections between this function and other existing topological notions are established and separate examples are given. It is shown that weakly semi-open functions on regular spaces are semi-open. We also introduce and study the concept of weakly semi-closed functions.

1. Introduction and Preliminaries. N. Levine [20] and N. Biswas [6], introduced and investigated the notions of semi-open sets and semiclosed sets in topological spaces, respectively. Since then, a lot of work has been done using these notions and many interesting results have been obtained [2, 6, 10, 13, 15, 26, 27]. D. A. Rose [28] and D. A. Rose with D. S. Janckovic [29] have defined the notions of weakly open and weakly closed functions and investigated some of the fundamental properties of these types of functions. In this paper, we introduce the notions of weakly semi-open and weakly semi-closed functions as a new generalization of weakly open and weakly closed functions, respectively. We investigate some properties of these functions comparing with the other related functions. In this connection, we obtain a new decomposition of semi-openness and closedness. It is also shown that a weakly semi-open inverse image of a semi-connected space is connected.

Throughout this paper, (X, τ) and (Y, σ) (or simply, X and Y) denote topological spaces on which no separation axioms are assumed unless explicitly stated. If S is any subset of a space X, then Cl(S) and Int(S) denote the closure and the interior of S, respectively. Recall that a set S is called regular open (respectively regular closed) if S = Int(Cl(S)) (respectively S = Cl(Int(S)). A point x ∈ X is called a θ-cluster [32] point of S if S ∩ Cl(U) ≠ ϕ for each open set U containing x. The set of all θ-cluster points of S is called the θ-closure of S and is denoted by Cl_{θ}(S). Hence, a subset S is called θ-closed [32] if Cl_{θ}(S) = S. The complement of a θ-closed set is called a θ-open set. The θ-interior of a subset S of X is the union of all open subsets of X whose closures are contained in S, and is denoted by Int_{θ}(S). A subset S ⊂ X is called semi-open [20] (respectively preopen [22], α-open [24], and β-open [1] (or semi-preopen [2])), if S ⊂ Cl(Int(S)) (respectively S ⊂ Int(Cl(S)), S ⊂ Int(Cl(Int(S)) and S ⊂ Cl(Int(Cl(S))). The complement of a semi-open set is called a semi-closed [6] set. The family of all semi-open (respectively semi-closed) sets of a space X is denoted by SO(X, τ) (respectively SC(X, τ)). The intersection of all semi-closed sets containing S is called the semi-closure of S [6,13] and is denoted by sCl(S). The semi-interior [13] of S is defined by the union of all semi-open sets contained in S and is denoted by sInt(S).

A space X is called extremally disconnected (E.D.) [33] if the closure of each open set in X is open. The space X is called semi-connected [27] if X cannot be expressed as the union of two nonempty disjoint semi-open sets.

A function f : (X, τ) → (Y, σ) is called:

(i) semi-continuous [20] if for each open subset V of Y, f^{−1}(V) ∈ SO(X, τ).

(ii) weakly semi-continuous [12, 17] if for each x ∈ X and each open subset V of Y containing f(x), there exists a semi-open subset U of X such that x ∈ U and f(U) ⊂ Cl(V).

(iii) semi-open [7] (respectively semi-closed [25]) if f(U) ∈ SO(Y, σ) (respectively f(U) ∈ SC(Y, σ)) for each open (respectively closed) subset U of X.

(iv) weakly open [28] if f(U) ⊂ Int(f(Cl(U))) for each open subset U of

X.

(v) weakly closed [29] if Cl(f(Int(F))) ⊂ f(F) for each closed subset F of X.

(vi) almost open in the sense of Singal and Singal, written as (a.o.S) [31] if

the image of each regular open subset U of X is an open set in Y. (vii) preopen [22] (respectively preclosed [14] , β-open [1], α-open [23]) if for

each open subset U (respectively closed subset F, open subset U, open subset U) of X, f(U) is a preopen (respectively f(F) is a preclosed, f(U) is a β-open, f(U) is an α-open) set in Y.

(viii) pre-semi-closed [16] if for each semi-closed subset B of X, f(B) is a

semi-closed in Y. (ix) contra-open [4] (respectively contra-closed [4]) if f(U) is closed (re-

spectively open) in Y for each open (respectively closed) subset U of X.

(x) completely continuous [3] if for each open subset V of Y, f^{−1}(V) is a regular open set in X.

(xi) complementary semi-weakly continuous [17] if for each open subset V of Y, f^{−1}(F r(V)) is semi-closed in X, where F r(V) denotes the frontier on V.

2. Weakly Semi-open Functions. Since semi-continuity [20] is dual to semi-openness [7], we define in this section the concept of weak semiopenness as a natural dual to the weak semi-continuity due to G. Costovici [12] and A. Kar et. al. [19].

\underline{Definition 2.1}. A function f: (X, τ) → (Y, σ) is said to be weakly semiopen if f(U) ⊂ sInt(f(Cl(U))) for each open subset U of X.

Clearly, every semi-open function is weakly semi-open, but the converse is not generally true.

\underline{Example 2.2}. A weakly semi-open function need not be semi-open. Let X = {a, b, c}, τ = {∅, {a}, {c}, {a, c}, X} and σ = {∅, {b}, {a, b}, {bc}, X}. Then the identity function f: (X, τ) → (X, σ) is weakly semi-open which is not semi-open, since for U = {a}, f(U) is not a semi-open set in (X, σ).

\underline{Theorem 2.3}. For a function f: (X, τ) → (Y, σ), the following conditions are equivalent:

(i) f is weakly semi-open. (ii) f(Int_{θ}(A)) ⊂ sInt(f(A)) for every subset A of X. (iii) Int_{θ}(f^{−1}(B)) ⊂ f^{−1}(sInt(B)) for every subset B of Y. (iv) f^{−1}(sCl(B)) ⊂ Cl_{θ}(f^{−1}(B)) for every subset B of Y.

\underline{Proof}.

(i) → (ii): Let A be any subset of X and x ∈ Int_{θ}(A). Then, there exists an open set U such that x ∈ U ⊂ Cl(U) ⊂ A. Then, f(x) ∈ f(U) ⊂ f(Cl(U)) ⊂ f(A). Since f is weakly semi-open, f(U) ⊂ sInt(f(Cl(U))) ⊂ sInt(f(A)). This implies that f(x) ∈ sInt(f(A)). This shows that x ∈ f^{−1}(sInt(f(A))). Thus, Int_{θ}(A) ⊂ f^{−1}(sInt(f(A))), and so, f(Int_{θ}(A)) ⊂ sInt(f(A)).

(ii) → (i): Let U be an open set in X. As U ⊂ Int_{θ}(Cl(U)) implies, f(U) ⊂ f(Int_{θ}(Cl(U))) ⊂ sInt(f(Cl(U))). Hence, f is weakly semi-open.

(ii) → (iii): Let B be any subset of Y. Then by (ii), f(Int_{θ}(f^{−1}(B)) ⊂ sInt(B). Therefore, Int_{θ}(f^{−1}(B)) ⊂ f^{−1}(sInt(B)).

(iii) → (ii): This is obvious.

(iii) → (iv): Let B be any subset of Y. Using (iii), we have X−Cl_{θ}(f^{−1}(B)) = Int_{θ}(X−f^{−1}(B)) = Int_{θ}(f^{−1}(Y −B)) ⊂ f^{−1}(sInt(Y − B)) = f^{−1}(Y − sCl(B)) = X − (f^{−1}(sCl(B)). Therefore, we obtain f^{−1}(sCl(B)) ⊂ Cl_{θ}(f^{−1}(B)).

(iv) → (iii): Similarly we obtain, X − f^{−1}(sInt(B)) ⊂ X − Int_{θ}(f^{−1}(B)), for every subset B of Y, i.e., Int_{θ}(f^{−1}(B)) ⊂ f^{−1}(sInt(B)).

Furthermore, we can prove the following theorem.

\underline{Theorem 2.4}. For a function f: (X, τ) → (Y, σ), the following statements are equivalent.

(i) f is weakly semi-open (ii) For each x ∈ X and each open subset U of X containing x, there exists

a semi-open set V containing f(x) such that V ⊂ f(Cl(U)).

\underline{Proof}.

(i) → (ii): Let x ∈ X and U be an open set in X with x ∈ U. Since f is weakly semi-open, f(x) ∈ f(U) ⊂ sInt(f(Cl(U))). Let V = sInt(f(Cl(U))). Hence, V ⊂ f(Cl(U)), with V containing f(x).

(ii) → (i): Let U be an open set in X and let y ∈ f(U). It follows from (ii) that V ⊂ f(Cl(U)) for some V semi-open in Y containing y. Hence, we have y ∈ V ⊂ sInt(f(Cl(U))). This shows that f(U) ⊂ sInt(f(Cl(U))), i.e., f is a weakly semi-open function.

\underline{Theorem 2.5}. Let f: (X, τ) → (Y, σ) be a bijective function. Then the following statements are equivalent.

(i) f is weakly semi-open, (ii) sCl(f(U)) ⊂ f(Cl(U)) for each U open in X, (iii) sCl(f(Int(F)) ⊂ f(F) for each F closed in X.

\underline{Proof}.

(i) → (iii): Let F be a closed set in X. Then we have f(X − F) = Y − f(F) ⊂ sInt(f(Cl(X − F))) and so Y − f(F) ⊂ Y − sCl(f(Int(F))). Hence, sCl(f(Int(F))) ⊂ f(F).

(iii) → (ii): Let U be an open set in X. Since Cl(U) is a closed set and U ⊂ Int(Cl(U)) by (iii) we have sCl(f(U)) ⊂ sCl(f(Int(Cl(U))) ⊂ f(Cl(U)).

(ii) → (iii): Similar to (iii) → (ii). (iii) → (i): Clear.

The proof of the following theorem is straightforward and thus is omitted.

\underline{Theorem 2.6}. For a function f: (X, τ) → (Y, σ) the following conditions are equivalent.

(i) f is weakly semi-open. (ii) f(Int(F)) ⊂ sInt(f(F)) for each closed subset F of X, (iii) f(Int(Cl(U))) ⊂ sInt(f(Cl(U))) for each open subset U of X, (iv) f(U) ⊂ sInt(f(Cl(U))) for each preopen subset U of X,

(v) f(U) ⊂ sInt(f(Cl(U))) for each α-open subset U of X.

Now, we recall a definition of strong continuity. This definition when combined with weak semi-openness implies semi-openness.

A function f: (X, τ) → (Y, σ) is said to be strongly continuous [22], if for every subset A of X, f(Cl(A)) ⊂ f(A).

\underline{Theorem 2.7}. If f: (X, τ) → (Y, σ) is weakly semi-open and strongly continuous, then f is semi-open.

\underline{Proof}. Let U be an open subset of X. Since f is weakly semiopen f(U) ⊂ sInt(f(Cl(U))). However, because f is strongly continuous, f(U) ⊂ sInt(f(U)) and therefore, f(U) is semi-open.

\underline{Example 2.8}. A semi-open function need not be strongly continuous. Let X = Y = {a, b, c}, τ = {∅, {a}, X} and σ = {∅, {a}, {a, b}, Y }. Then the identity function f: (X, τ) → (Y, σ) is semi-open but is not strongly continuous since, if U = {a}, f(Cl(U)) = f(X) = X ⊄ f(U).

Recall that f: (X, τ) → (Y, σ) is said to be relatively weakly open [5] provided that f(U) is open in f(Cl(U)) for every open subset U of X.

\underline{Theorem 2.9}. A function f: (X, τ) → (Y, σ) is semi-open if f is weakly semi-open and relatively weakly open.

\underline{Proof}. Assume f is weakly semi-open and relatively weakly open. Let U be an open subset of X and let y ∈ f(U). Since f is relatively weakly open there is an open subset V of Y for which f(U) = f(Cl(U)) ∩ V. Because f is weakly semi-open, it follows that f(U) ⊂ sInt(f(Cl(U))). Then y ∈ sInt(f(Cl(U))) ∩ V ⊂ f(Cl(U)) ∩ V = f(U) and therefore, f(U) is semi-open.

\underline{Theorem 2.10}.

(i) If f: (X, τ) → (Y, σ) is contra-closed, then f is a weakly semi-open function.

(ii) If f: (X, τ) → (Y, σ) is preopen and contra-open, then f is weakly semi-open.

\underline{Proof}.

(i) Let U be an open subset of X. Then, we have f(U) ⊂ f(Cl(U))) = sInt(f(Cl(U))).

(ii) Let U be an open subset of X. Since f is preopen f(U) ⊂ Int(Cl(f(U))) and since f is contra-open f(U) is closed. Therefore, f(U) ⊂ Int(Cl(f(U))) = Int(f(U)) ⊂ Int(f(Cl(U))) ⊂ sInt(f(Cl(U))).

The converse of Theorem 2.10 does not hold.

\underline{Example 2.11}. A weakly semi-open function need not be contra-closed is given from Example 2.8.

\underline{Theorem 2.12}. Let X be a regular space. Then f: (X, τ) → (Y, σ) is weakly semi-open if and only if f is semi-open.

\underline{Proof}. The sufficiency is clear. For the necessity, let W be a nonempty open subset of X. For each x in W, let U_{x} be an open set such that x ∈ U_{x} ⊂ Cl(U_{x}) ⊂ W. Hence, we obtain that W = ∪{U_{x} : x ∈ W } = ∪{Cl(U_{x}) : x ∈ W } and, f(W) = ∪{f(U_{x}) : x ∈ W } ⊂ ∪{sInt(f(Cl(U_{x}))) : x ∈ W } ⊂ sInt(f(∪{Cl(U_{x}) : x ∈ W }) = sInt(f(W)). Thus, f is semi-open.

Next, we define the dual form of a complementary semi-weakly continuous function called a complementary weakly semi-open function.

\underline{Definition 2.13}. A function f: (X, τ) → (Y, σ) is called complementary weakly semi-open (written as c.w.s.o.) if for each open set U of X, f(F r(U)) is semi-closed in Y, where F r(U) denotes the frontier of U.

The following two examples show that weakly semi-open and c.w.s.o. are independent of each other.

\underline{Example 2.14}. A weakly semi-open function need not be c.w.s.o. Let X = {a, b}, τ = {∅, {b}, X}, Y = {x, y} and σ = {∅, {x}, Y }. Let f: (X, τ) → (Y, σ) be given by f(a) = x and f(b) = y. Then f is clearly weakly semi-open, but is not c.w.s.o., since F_{r}({b}) = Cl({b}) − {b} = {a} and f(F_{r}({b})) = {x} is not a semi-closed set in Y.

\underline{Example 2.15}. c.w.s.o. does not imply weakly semi-open. Let X = {a, b}, τ = {∅, {a}, {b}, X}, Y = {x, y} and σ = {∅, {y}, Y }. Let f: (X, τ) → (Y, σ) be given by f(a) = x and f(b) = y. Then f is not weakly semi-open, but f is c.w.s.o., since the frontier of every open set is the empty set and f(∅) = ∅ is semi-closed.

\underline{Theorem 2.16}. Let Y be an extremally disconnected space. If f: (X, τ) → (Y, σ) is bijective weakly semi-open and c.w.s.o., then f is semi-open.

\underline{Proof}. Let U be an open subset in X with x ∈ U, since f is weakly semi-open, by Theorem 2.4 there exists a semi-open set V containing f(x) = y such that V ⊂ f(Cl(U)). Now F r(U) = Cl(U) − U and thus x ∉ F r(U). Hence, y ∉ f(F r(U)) and therefore, y ∈ V − f(F r(U)). Put V_{y} = V − f(F r(U)), a semi-open set since f is c.w.s.o and Y is extremally disconnected. Since y ∈ V_{y}, y ∈ f(Cl(U)). But y ∉ f(F r(U)) and thus, y ∉ f(F r(U)) = f(Cl(U)) − f(U) which implies that y ∈ f(U). Therefore, f(U) = ∪{V_{y} : V_{y} ∈ SO(Y, σ), y ∈ f(U)}. Hence, f is semi-open.

The following theorem is a variation of a result of C. Baker [4] in which contra-openness is replaced with weakly semi-open and closed by contra-pre-semi-closed, where, f: (X, τ) → (Y, σ) is said to be contra-pre-semi-closed [9] provided that f(F) is semi-open for each semi-closed subset F of X.

\underline{Theorem 2.17}. If f: (X, τ) → (Y, σ) is weakly semi-open, SC(Y, σ) is closed under arbitrary unions and if for each semi-closed subset F of X and each fiber f^{−1}(y) ⊂ X − F there exists an open subset U of X for which F ⊂ U and f^{−1}(y) ∩ Cl(U) = ϕ, then f is contra-pre-semi-closed.

\underline{Proof}. Assume F is a semi-closed subset of X and let y ∈ Y − f(F). Thus, f^{−1}(y) ⊂ X − F and hence, there exists an open subset U of X for which F ⊂ U and f^{−1}(y) ∩ Cl(U) = ϕ. Therefore, y ∈ Y − f(Cl(U)) ⊂ Y − f(F). Since f is weakly semi-open f(U) ⊂ sInt(f(Cl(U))). By complement, we obtain y ∈ sCl(Y − f(Cl(U))) ⊂ Y − f(F). Let B_{y} = sCl(Y − f(Cl(U))). Then B_{y} is a semi-closed subset of Y containing y. Hence, Y − f(F) = ∪{B_{y} : y ∈ Y − f(F)} is semi-closed and therefore, f(F) is semi-open.

\underline{Theorem 2.18}. If f: (X, τ) → (Y, σ) is an a.o.S function, then it is a weakly semi-open function.

\underline{Proof}. Let U be an open set in X. Since f is a.o.S and Int(Cl(U)) is regular open, f(Int(Cl(U))) is open in Y and hence, f(U) ⊂ f(Int(Cl(U)) ⊂ Int(f(Cl(U))) ⊂ sInt(f(Cl(U))). This shows that f is weakly semi-open.

The converse of Theorem 2.18 is not true in general.

\underline{Example 2.19}. A weakly semi-open function need not be a.o.S. Let X = Y = {a, b, c}, τ = {∅, {a}, {c}, {a, c}, X}, and σ = {∅, {b}, {a, b}, {b, c}, X}. Let f: (X, τ) → (Y, σ) be the identity. Then f is not a.o.S since Int(f(Int(Cl({a})))) = ∅. But f is weakly semi-open.

\underline{Lemma 2.20}. If f: (X, τ) → (Y, σ) is a continuous function, then for any subset U of X, f(Cl(U)) ⊂ Cl(f(U)) [28].

\underline{Theorem 2.21}. If f : (X, τ) → (Y, σ) is a weakly semi-open and continuous function, then f is a β-open function.

\underline{Proof}. Let U be an open set in X. Then by weak semi-openness of f, f(U) ⊂ sInt(f(Cl(U))). Since f is continuous f(Cl(U)) ⊂ Cl(f(U)). Hence, we obtain that f(U) ⊂ sInt(f(Cl(U))) ⊂ sInt(Cl(f(U))) ⊂ Cl(Int(Cl(f(U)))). Therefore, f(U) ⊂ Cl(Int(Cl(f(U)))) which shows that f(U) is a β-open set in Y. Thus, f is a β-open function.

Since every strongly continuous function is continuous we have the following corollary.

\underline{Corollary 2.22}. If f: (X, τ) → (Y, σ) is a weakly semi-open and strongly continuous function, then f is a β-open function.

\underline{Theorem 2.23}. If f: (X, τ) → (Y, σ) is an injective weakly semi-open function of a space X onto a semi-connected space Y, then X is connected.

\underline{Proof}. Let us assume that X is not connected. Then there exist nonempty open sets U_{1} and U_{2} such that U_{1} ∩ U_{2} = ϕ and U_{1} ∪ U_{2} = X. Hence, we have f(U_{1}) ∩ f(U_{2}) = ϕ and f(U_{1}) ∪ f(U_{2}) = Y. Since f is weakly semi-open, we have f(U_{i}) ⊂ sInt(f(Cl(U_{i}))) for i = 1, 2, and since U_{i} is open and also closed, we have f(Cl(U_{i})) = f(U_{i}) for i = 1, 2. Hence, f(U_{i}) is semi-open in Y for i = 1, 2. Thus, Y has been decomposed into two nonempty disjoint semi-open sets. This is contrary to the hypothesis that Y is a semi-connected space. Thus, X is connected.

\underline{Definition 2.24}. A space X is said to be hyperconnected [26] if every nonempty open subset of X is dense in X.

\underline{Theorem 2.25}. If X is a hyperconnected space, then a function f: (X, τ) → (Y, σ) is weakly semi-open if and only if f(X) is semi-open in Y.

\underline{Proof}. The sufficiency is clear. For the necessity observe that for any open subset U of X, f(U) ⊂ f(X) = sInt(f(X)) = sInt(f(Cl(U))).

3. Weakly semi-closed Functions. Now, we define a generalized form of semi-closed functions.

\underline{Definition 3.1}. A function f: (X, τ) → (Y, σ) is said to be weakly semiclosed if sCl(f(Int(F))) ⊂ f(F) for each closed set F in X.

Clearly,

(i) Every closed function is α-closed and every α-closed function is semiclosed, but the reverse implications are not true in general [23].

(ii) every semi-closed function is a weakly semi-closed function, but the converse is not generally true, as the next example shows.

\underline{Example 3.2}.

(i) Let X = {x, y, z} and τ = {∅, {x}, {x, y}, X}. Then a function f: (X, τ) → (X, τ) which is defined by f(x) = x, f(y) = z and f(z) = y is α-open and α-closed but neither open nor closed [23].

(ii) Let f: (X, τ) → (Y, σ) be the function from Example 2.2. Then f is weakly semi-closed but not semi-closed.

From the observation above and Example 3.2, we have the following diagram.

closed (open) func.  ⟶ α-closed (α-open) func.
↓  ↓
weakly semi-closed (semi-open) func. ⟵ semi-closed (semi-open) func.

\underline{Theorem 3.3}. For a function f: (X, τ) → (Y, σ), the following conditions are equivalent.

(i) f is weakly semi-closed. (ii) sCl(f(U)) ⊂ f(Cl(U)) for every open subset U of X.

\underline{Proof}.

(i) → (ii). Let U be any open subset of X. Then sCl(f(U)) = sCl(f(Int(U))) ⊂ sCl(f(Int(Cl(U)))) ⊂ f(Cl(U)).

(ii) → (i). Let F be any closed subset of X. Then, sCl(f(Int(F))) ⊂ f(Cl(Int(F))) ⊂ f(Cl(F)) = f(F).

The proof of the following theorem is straightforward and is therefore omitted.

\underline{Theorem 3.4}. For a function f: (X, τ) → (Y, σ) the following conditions are equivalent:

(i) f is weakly semi-closed, (ii) sCl(f(U)) ⊂ f(Cl(U)) for each open subset U of X, (iii) sCl(f(Int(F))) ⊂ f(Cl(Int(F))) for each closed subset F of X, (iv) sCl(f(Int(F))) ⊂ f(Cl(Int(F))) for each regular closed subset F of

X,

(v) sCl(f(Int(F))) ⊂ f(F) for each pre-closed subset F of X, (vi) sCl(f(Int(F))) ⊂ f(F) for every α-closed subset F of X.

\underline{Remark 3.5}. By Theorem 2.5, if f: (X, τ) → (Y, σ) is a bijective function, then f is weakly semi-open if and only if f is weakly semi-closed.

\underline{Theorem 3.6}.

(i) If f: (X, τ) → (Y, σ) is contra-open, then f is weakly semi-closed. (ii) If f: (X, τ) → (Y, σ) is preclosed and contra-closed, then f is weakly

semi-closed.

\underline{Proof}.

(i) Let F be a closed subset of X. Then sCl(f(Int(F))) ⊂ f(Int(F)) ⊂ f(F).

(ii) Let F be a closed subset of X. Since f is preclosed Cl(Int(f(F))) ⊂ f(F) and since f is contra-closed f(F) is open. Therefore, sCl(f(Int(F))) ⊂ sCl(f(F)) ⊂ Cl(Int(f(F))) ⊂ f(F).

G. L. Garg and D. Sivaraj in [16] showed that if f is a pre-semi-closed function then sCl(f(A)) ⊂ f(sCl(A)) for every subset A of X. Therefore, every pre- semi-closed function is weakly semi-closed.

\underline{Theorem 3.7}. If f: (X, τ) → (Y, σ) is one-to-one and weakly semiclosed, then for every subset F of Y and every open set U in X with f^{−1}(F) ⊂ U, there exists a semi-closed set B in Y such that F ⊂ B and f^{−1}(B) ⊂ Cl(U).

\underline{Proof}. Let F be a subset of Y and let U be an open subset of X with f^{−1}(F) ⊂ U. Put B = sCl(f(Int(Cl(U)))), then B is a semiclosed subset of Y such that F ⊂ B since F ⊂ f(U) ⊂ f(Int(Cl(U))) ⊂ sCl(f(Int(Cl(U)))) = B. And since f is weakly semi-closed, f^{−1}(B) ⊂ Cl(U).

Taking the set F in Theorem 3.7 to be y for y ∈ Y we obtain the following result.

\underline{Corollary 3.8}. If f: (X, τ) → (Y, σ) is one-to-one and weakly semiclosed, then for every point y in Y and every open set U in X with f^{−1}(y) ⊂ U, there exists a semi-closed set B in Y containing y such that f^{−1}(B) ⊂ Cl(U).

Recall, that two nonempty sets A and B of X are strongly separated [29], if there exist open sets U and V in X with A ⊂ U and B ⊂ V and Cl(U) ∩ Cl(V) = ϕ. If A and B are singleton sets we may speak of points being strongly separated. We will use the fact that in a normal space, disjoint closed sets are strongly separated.

A space X is said to be Ultra semi-Hausdorff or in short Ultra semi-T_{2} if for every pair of distinct points x and y, there exist two semi-closed sets U and V such that x ∈ U and y ∈ V and U ∩ V = ϕ.

\underline{Theorem 3.9}. If f: (X, τ) → (Y, σ) is a weakly semi-closed surjection and all pairs of disjoint fibers are strongly separated, then Y is Ultra semi-T_{2}.

\underline{Proof}. Let y and z be two points in Y. Let G and H be open sets in X such that f^{−1}(y) ∈ G and f^{−1}(z) ∈ H with Cl(G) ∩ Cl(H) = ϕ. By weak semi-closedness (Theorem 3.7) there are semi-closed sets U and V in Y such that y ∈ U and z ∈ V, f^{−1}(U) ⊂ Cl(G) and f^{−1}(V) ⊂ Cl(H). Therefore, U ∩ V = ϕ because Cl(G) ∩ Cl(H) = ϕ and f surjective. Then Y is Ultra semi-T_{2}.

\underline{Corollary 3.10}. If f: (X, τ) → (Y, σ) is a weakly semi-closed surjection with all fibers closed and X is normal, then Y is Ultra semi-T_{2}.

\underline{Corollary 3.11}. If f: (X, τ) → (Y, σ) is a continuous weakly semi-closed surjection with X a compact T_{2} space and Y a T_{1} space, then Y is a compact Ultra semi-T_{2} space.

\underline{Proof}. Since f is a continuous surjection and Y is a T_{1} space, Y is compact and all fibers are closed. Since X is normal Y is also Ultra semi-T_{2} space.

\underline{Definition 3.12}. A topological space X is said to be quasi H-closed [11] (respectively SC-closed [8]), if every open (respectively semi-closed) cover of X has a finite subfamily whose closures cover X. A subset A of a topological space X is quasi H-closed relative to X (respectively SC-closed relative to X) if every cover of A by open (respectively semi-closed) subsets of X has a finite subfamily whose closures cover A.

\underline{Remark 3.13}. D. S. Jankovic in [18] showed that a space X is extremally disconnected if and only if the intersection of two semi-open sets is semi-open, consequently the union of two semi-closed sets is semi-closed.

\underline{Lemma 3.14}. A function f: (X, τ) → (Y, σ) is open if and only if for each B ⊂ Y, f^{−1}(Cl(B)) ⊂ Cl(f^{−1}(B)) [30].

\underline{Theorem 3.15}. Let X be an extremally disconnected space. Let f: (X, τ) → (Y, σ) be an open, weakly semi-closed function which is one-to-one and such that f^{−1}(y) is quasi H-closed relative to X for each y in Y. If G is SC-closed relative to Y then f^{−1}(G) is quasi H-closed.

\underline{Proof}. Let \{V_{β} : β ∈ I\}, I being the index set, be an open cover of f^{−1}(G). Then for each y ∈ G∩f(X), f^{−1}(y) ⊂ ∪{Cl(V_{β}) : β ∈ I(y)} = H_{y} for some finite subfamily I(y) of I. Since X is extremally disconnected each Cl(V_{β}) is open, hence, H_{y} is open in X. So by Corollary 3.8, there exists a semi-closed set U_{y} containing y such that f^{−1}(U_{y}) ⊂ Cl(H_{y}). By Remark 3.13 {U_{y} : y ∈ G ∩ f(X)} ∪ {Y − f(X)} is a semi-closed cover of G, G ⊂ ∪{Cl(U_{y}) : y ∈ K} ∪ {Cl(Y − f(X))} for some finite subset K of g ∩ f(X). Hence and by Lemma 3.14, f^{−1}(G) ⊂ ∪{f^{−1}(Cl(U_{y})) : y ∈ K} ∪ {f^{−1}(Cl(Y − f(X)))} ⊂ ∪{Cl(f^{−1}(U_{y})) : y ∈ K}∪ {Cl(f^{−1}(Y − f(X)))} ⊂ {Cl(f^{−1}(U_{y})) : y ∈ K}, so f^{−1}(G) ⊂ ∪{Cl(V_{β}) : β ∈ I(y), y ∈ K}. Therefore, f^{−1}(G) is quasi H-closed.

\underline{Corollary 3.16}. Let f: (X, τ) → (Y, σ) be as in Theorem 3.15. If Y is SC-closed, then X is quasi H-closed.

\underline{Acknowledgment}. The authors are grateful to the referee for his remarkable work which improved the quality of this paper.

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Mathematics Subject Classification (2000): 54A40, 54C10, 54D10, 54C08

Miguel Caldas Departamento de Matematica Aplicada Universidade Federal Fluminense Rua Mario Santos Braga, s/n CEP: 24020-140 Niteroi, RJ. Brasil email: gmamccs@vm.uff.br

G. B. Navalagi Department of Mathematics KLE Societys G. H. College, Haveri-581110, Karnataka, India e-mail: gnavalagi@hotmail.com