A NOTE ON ALMOST CONTRA-PRECONTINUOUS FUNCTIONS

Almost contra-precontinuous functions were introduced by Ekici [7] and recently have been investigated further by Noiri and Popa [13]. The purpose of this note is to develop some new characterizations of almost contra-precontinuous functions and to introduce a new weak form of almost contra-precontinuity, which we call subalmost contraprecontinuity. It is shown that subalmost contra-precontinuity implies subalmost weak continuity and is independent of subweak continuity. Subalmost contra-precontinuity is used to extend several results in the literature concerning almost contra-precontinuity. For example, we show that the graph of a subalmost contra-precontinuous function with a Hausdorff codomain is P-regular and that the domain of a subalmost contraprecontinuous injection with a weakly Hausdorff codomain is pre-T1. These results extend the analogous results for an almost contra-precontinuous function.


Introduction
Almost contra-precontinuous functions were introduced by Ekici [7] and recently have been investigated further by Noiri and Popa [13].The purpose of this note is to develop some new characterizations of almost contra-precontinuous functions and to introduce a new weak form of almost contra-precontinuity, which we call subalmost contraprecontinuity.It is shown that subalmost contra-precontinuity implies subalmost weak continuity and is independent of subweak continuity.Subalmost contra-precontinuity is used to extend several results in the literature concerning almost contra-precontinuity.For example, we show that the graph of a subalmost contra-precontinuous function with a Hausdorff codomain is P-regular and that the domain of a subalmost contraprecontinuous injection with a weakly Hausdorff codomain is pre-T 1 .These results extend the analogous results for an almost contra-precontinuous function.

Preliminaries
The symbols X and Y denote topological spaces with no separation axioms assumed unless explicitly stated.All sets are considered to be subsets of topological spaces.The closure and interior of a set A are signified by Cl(A) and Int(A), respectively.A set A is regular open if A = Int(Cl(A)).A set A is preopen [12] (resp., semiopen [11], β-open [1]) provided that The preclosure [8] of A, denoted by p Cl(A), is the intersection of all preclosed sets containing A. The semiclosure [5] of a set A denoted by sCl(A), and β-closure [2] of a set A denoted by β Cl(A) are defined analogously.The θ-semi-closure [9] of a subset A of a space X, denoted by sCl θ (A), is the set of all x ∈ X such that Cl(V ) ∩ A = ∅ for every semiopen subset V of X containing x.The set of all preopen subsets of a space X is denoted by PO(X) and the collection of all preopen subsets of X containing a fixed point x is denoted by PO(X,x).The sets SO(X), SO(X,x), βO(X), βO(X,x), PC(X), and RO(X) are defined analogously.Finally, if an operator is used with respect to a proper subspace, then a subscript will be added to the operator.Otherwise, it is assumed that the operator refers to the space X or Y .
Definition 2.2.A function f : X → Y is said to be subweakly continuous [14] (resp., subalmost weakly continuous [3], subweakly β-continuous [4]) provided that there is an open base Ꮾ for the topology on Y such that for every for every open subset V of Y .

Almost contra-precontinuous functions
Noiri and Popa proved the following characterizations of almost contra-precontinuity. Theorem 3.1 (Noiri and Popa [13]).For a function f : X → Y , the following properties are equivalent: (a) f is almost contra-precontinuous; We extend these characterizations by showing that Theorem 3.1(c) can be stated for open sets only.The following lemmas will be useful.
The next result is an immediate consequence of Theorems 3.1 and 3.4.

be a function and let be any collection of subsets of Y containing the open sets. Then f is almost contra-precontinuous if and only if p
Corollary 3.6.For a function f : X → Y , the following properties are equivalent:

Subalmost contra-precontinuous functions
We define a function f : X → Y to be subalmost contra-precontinuous provided that there exists an open base Ꮾ for the topology on Y such that p Cl( for every V ∈ Ꮾ. Obviously almost contra-precontinuity implies subalmost contraprecontinuity.The following example shows that the converse does not hold. Recall that a space X is extremally disconnected (ED) if the closure of every open set is open in X.
Example 4.1.Let X be a non-ED, T 1 -space and let Y = X have the discrete topology.The identity mapping f : X → Y is subalmost contra-precontinuous with respect to the base for Y consisting of the singleton sets.However, f is not almost contra-precontinuous.

Note that for y
Since sCl(A) ⊆ Cl(A) for every set A, it follows that subalmost contra-precontinuity implies subalmost weak continuity, and hence it also implies subweak β-continuity.The following example shows that subalmost contra-precontinuity and subalmost weak continuity are not equivalent.
Since the function in Example 4.2 is obviously subweakly continuous, we see that subweak continuity does not imply subalmost contra-precontinuity.The following example 4 Almost contra-precontinuous functions completes the proof that subalmost contra-precontinuity is independent of subweak continuity.
Example 4.3.Let X be an indiscrete space with at least two points and let Y = X have the discrete topology.Since p Cl({x}) = {x} for very x ∈ X, the identity mapping f : X → Y is subalmost contra-precontinuous with respect to the base for Y consisting of the singleton sets.However, since every singleton set of X is dense, f is not subweakly continuous.
The following characterizations of subalmost contra-precontinuity are analogous to those in Theorem 3.4 for almost contra-precontinuity.
Since subweak continuity implies subalmost weak continuity, we have the following result.

Graph-related properties of subalmost contra-precontinuous functions
By the graph of a function f : X → Y , we mean the subset G( f ) = {(x, f (x)) : x ∈ X} of the product space X × Y .Definition 5.1.The graph of a function f : X → Y , G( f ) is said to be P-regular [7] provided that for every (x, y) ∈ X × Y − G( f ), there exist a preclosed subset U of X and regular open subset Recall that the graph function of a function f : X → Y is the function g : X → X × Y given by g(x) = (x, f (x)) for every x ∈ X. Theorem 5.4.Let f : (X,τ) → (Y ,σ) be a function and let Ꮾ be an open base for σ.Let If we let Ꮾ = σ in Theorem 5.4, we obtain the following result.
Corollary 5.5.If the graph function of f : X → Y is subalmost contra-precontinuous with respect to the usual base for the product topology for the product space X × Y , then f is almost contra-precontinuous.
Corollary 5.6 (Ekici [7,Theorem 4]).If the graph function of f : Recall that a space X is said to be zero-dimensional provided that X has a clopen base.
Theorem 5.7.If the function f : X → Y is subalmost contra-precontinuous and X is zero-dimensional, then the graph function of f , g : ). Therefore the graph function g is subalmost contra-precontinuous.
Remark 5.8.In Theorem 5.7 the requirement that X be zero-dimensional can be replaced by the assumption that X is an ED space.

Additional properties of subalmost contra-precontinuous functions
The following generalizations of the T 1 and Hausdorff properties will be useful.
6 Almost contra-precontinuous functions Definition 6.1.A space X is said to be pre-T 1 [10] provided that for every pair of distinct points x and y of X, there exist preopen sets U and V containing x and y, respectively, with y / ∈ U and x / ∈ V .Definition 6.2.A space X is said to be weakly Hausdorff [15] if each element of X is an intersection of regular closed sets.
Theorem 6.3.If f : X → Y is a subalmost contra-precontinuous injection and Y is weakly Hausdorff, then X is pre-T 1 .
Proof.Let x 1 and x 2 be distinct points in X.Then f (x 1 ) = f (x 2 ), and since Y is weakly Hausdorff, there exists a regular closed subset , which is preopen.Also f (x 1 ) ∈ F, which is regular closed and therefore also semiopen.Since F ∩ V = ∅, it follows that f (x 1 ) / ∈ sCl(V ), and hence Corollary 6.4 (Ekici [7,Theorem 11]).If f : X → Y is an almost contra-precontinuous injection and Y is weakly Hausdorff, then X is pre-T 1 .
The following example shows that the restriction of a subalmost contra-precontinuous function is not necessarily subalmost contra-precontinuous. Example 6.5.Let X = {a, b,c,d} have the topology τ = {X, ∅, {a, b}} and let Y = X have the discrete topology.Since the singleton subsets of X are preclosed [10], the identity mapping f : X → Y is subalmost contra-precontinuous with respect to the base for Y consisting of the singleton sets.However, if A = {a, c}, then f | A : A → Y fails to be subalmost contra-precontinuous.
Next we show that the restriction of a subalmost contra-precontinuous function to a semiopen set is subalmost contra-precontinuous.The following lemma will be useful.Lemma 6.6 (Baker [3]).If B ⊆ A ⊆ X and A is semiopen in X, then p Cl A (B) ⊆ p Cl(B).Proof.Let V∈Ꮾ.Then using Lemma 6.6, we see that p Cl A ( f A (sCl(V )).Hence, f | A : A → Y is subalmost contra-precontinuous with respect to Ꮾ.
If we take Ꮾ to be the topology on Y in Theorem 6.7, we obtain the following result.Corollary 6.8 (Ekici [7,Theorem 2]).If f : X → Y is almost contra-precontinuous and A is a semiopen subset of X, then f | A : A → Y is almost contra-precontinuous.

Theorem 4 . 4 .
For a function f : X → Y , the following conditions are equivalent: (a) f is subalmost contra-precontinuous; (b) there exists an open base Ꮾ for Y such that p Cl

Corollary 4 . 6 .
If f : X → Y is subweakly continuous and satisfies the additional property that images of preclosed sets are open, then f is subalmost contra-precontinuous. Theorem 4.7.If f : X → Y is subalmost contra-precontinuous and semicontinuous, then f is subweakly continuous.Proof.Since f is subalmost contra-precontinuous, there exists an open base Ꮾ for the topology on Y such that p Cl

Theorem 6 . 7 .
If f : X → Y is subalmost contra-precontinuous with respect to the open base Ꮾ for Y and A is a semiopen subset of X, then f | A : A → Y is subalmost contraprecontinuous with respect to Ꮾ.