A Generalization of A Contra Pre Semi-Open Maps

The concept of θ-semi-open sets in topological spaces was introduced in 1984 and 1986 by T. Noiri [9, 10]. In this paper we introduce and study a generalization of a contra pre semi-open maps due to (Caldas and Baker) [3], it is called contra pre θs-open maps, the maps whose images of a θ-semi-open sets is θ-semi-closed. Also, we introduce and study a new type of closed maps called contra pre θs-closed maps, which is stronger than contra pre semi-closed due to Caldas [2], the maps whose image of a θsemi-closed sets is θ-semi-open.1991 Math. Subject Classification: 54 C10, 54 D 10.


Introduction
The concept of -semi-open set in topological spaces was introduced in 1984 and 1986 by T. Noiri [9,10], which depends on semi-open sets due to N. Levine [8]. When semi-open sets are replaced by -semi-open sets, new results are obtained. M. Caldas and C. Baker defined and studied the concept of contra pre semi-open maps [3], where the maps whose images of semi-open sets are semi-closed.
In this direction we shall define the concept of Pre s-open maps. In this paper we introduce two new types of open and closed maps called contra pre s-open and contra pre s-closed maps via the concept of semi-open sets and study some of their basic properties. We also establish relationships a mong these maps with other types of continuity, openness and closedness.

Preliminaries
Throughout the present paper, spaces always mean topological spaces on which no separation axiom is assumed unless explicitly stated. Let S be a subset of a space X. The closure and the interior of S are denoted by Cl(S) and Int(S), respectively. A subset S is said to be regular open(resp. semi-open [8]) if S = Int(Cl(S)) (resp. S  Cl(Int((S))). A subset S is said to be -semi-open [9] if for each x  S, there exists a semi-open set U in X such that x  U  Cl(U)  S. The complement of each regular open (resp. semi-open and -semi-open) set is called regular closed (resp. semi-closed and -semi-closed). The family of all semi-open (resp. semi-closed, -semiopen and -semi-closed) sets of X is denoted by SO(X) (resp. SC(X), SO(X) and SC(X)). A point x is said to be in the -semi-closure [10] [3] (resp. contra pre semi-closed [2]) if for each semi-open (resp. semiclosed) set U of X, f (U)  SC(Y) (resp. f (U)  SO(Y)).

Contra pre s-open and contra pre s-closed maps
Let f : (X, ) →(Y, ) be a map from a topological space (X, ) into a topological space (Y, ).
The proof of the following two lemmas follows directly from their definitions and, therefore, they are omitted.

Lemma 3.2:
Every contra pre s-closed map is contra pre semi-closed.
The converse of the above lemmas is not true in general as it is shown by the following examples.
Then f is contra pre semi-closed, but it is not contra pre s-closed.  ii) for every subset D of Y and for every -semi-closed subset G of X with iii) for every y Y and for every -semi-closed subset G of X with     therefore, y = f (x) for some x  G and we have x  f -1 (H)  A = X \ G which is a contradiction. Since D = H, that is, Y \ f (G) = H, which implies that f (G) is -semi-open and hence f is contra pre s-closed.

Proof: (i)(ii). Let D be a subset of Y and let G be a
Taking the set D in Theorem 3.2 to be {y} for y  Y we obtain the following result.

(d)(a). Let A be any -semi-open subset of X and set
The proof of the following theorem is similar to the above theorem for the contra pre s-closed maps.

Theorem 3.6:
For a map f : X → Y, the following are equivalent: ii) Since f is contra pre s-closed and since sCl (A) is -semi-closed, then f (A)  f (sCl (A)) = sInt ( f (sCl (A))) for every subset A of X.
Recall, that a map f : (X, ) → (Y, ) is called S-closed [4] if sCl ( f (A))  f (sCl (A)) for every subset A of X.

Lemma 3.3[7]: If Y is a regular closed subset of a space X and A  Y, then A is -semi-open in X if and only if A is -semi-open in Y.
Regarding the restriction f | R of a map f : (X, ) → (Y, ) to a subset R of X we have the following: The proof of the following result is not hard, therefore, it is omitted. a) If f is S-continuous surjection, then g is contra pre s-closed. b) If g is S-continuous injection, then f is contra pre s-closed.
Proof: a) Suppose G is any arbitrary -semi-closed set in Y. Since f is Scontinuous. Therefore, by [10, Theorem 1.1], f -1 (G) is -semi-closed in X.
Since g o f is contra pre s-closed and f is surjective (g  f ) ( f -1 (G)) = g (G) is -semi-open in Z. This implies that g is a contra pre s-closed map. b) Suppose G is any arbitrary -semi-closed set in X. Since g o f is contra This implies that f is a contra pre s-closed map.
Arguing as in the proof of Theorem 3.11, we obtain the following result.

Definition 3.3[7]:
A space X is said to be strongly semi-T2 if and only if for each two distinct points x and y in X, there exists two disjoint -semi-open sets A and B in X containing x and y, respectively. Theorem 3.14: If X is a strongly semi-T2 space and f :X→Y is contra pre s-open map, then the set A = {(x1, x2) : f (x1) = f (x2)} is -semi-closed in the product space X  X.