A NOTE ON θ-GENERALIZED CLOSED SETS

The purpose of this note is to strengthen several results in the literature concerning the preservation of θ-generalized closed sets. Also conditions are established under which images and inverse images of arbitrary sets are θ-generalized closed. In this process several new weak forms of continuous functions and closed functions are developed. 2000 Mathematics Subject Classification. Primary 54C10.


Introduction. Recently Dontchev and Maki
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 of θ-generalized closed sets.This is done by using the notion of a θ-c-closed set developed by Baker [2].These sets turn out to be a very natural tool to use in investigating the preservation of θ-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, called θ-g-c-continuous and θ-g-c-closed, respectively.It is shown that θ-g-c-continuity is strictly weaker than strong θ-continuity and that θ-g-c-closed is strictly weaker than θ-g-closed.

Preliminaries.
The symbols X and Y denote topological spaces with no separation axioms assumed unless explicitly stated.If A is a subset of a space X, then the closure and interior of A are denoted by Cl(A) and Int(A), respectively.The θ-closure of A [8], denoted by Cl θ (A), is the set of all x ∈ X for which every closed neighborhood of The following theorem from [5] gives a useful characterization of θ-g-openness.

Theorem 2.2 (Dontchev and Maki [5]). A set A is θ-g-open if and only if F ⊆ Int θ (A)
whenever F ⊆ A and F is closed.

Definition 2.3 (Dontchev and Maki [5]). A function
Definition 2.4 (Dontchev and Maki [5]).A function f : Definition 2.5 (Noiri [7]).A function f : X → Y is said to be strongly θ-continuous provided that, for every x ∈ X and every open neighborhood V of f (x), there exists an open neighborhood U of x for which f (Cl(U)) ⊆ V .[5] proved that the θ-g-closed, continuous image of a θ-g-closed set is θ-gclosed.In this section, we strengthen this result by replacing both the θ-g-closed and continuous requirements with weaker conditions.Our replacement for the θ-g-closed condition uses the concept of a θ-c-open set from [2].Definition 3.1 (Baker [2]).A set A is said to be θ-c-closed provided there is a set B for which A = Cl θ (B).

We define a function
Since θ-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.
Example 3.2.Let X = {a, b, c} have the topology τ = {X, ∅, {a}, {a, b}, {a, c}} and let f : X → X be the identity mapping.Since the θ-closure of every nonempty set is X, f is obviously θ-g-c-closed.However, since f ({c}) fails to be θ-g-closed, f is not θ-g-closed.
Corollary 3.4 (Dontchev and Maki [5]).If f : X → Y is continuous and θ-g-closed, then f (A) is θ-g-closed in Y for every θ-g-closed subset A of X.

Theorem 3 .
3 can be strengthened further by replacing continuity with a weaker condition.Instead of requiring inverse images of open sets to be open, we require that the inverse images of open sets interact with θ-g-closed sets in the same way as open sets.