© Hindawi Publishing Corp. RC-CONTINUOUS FUNCTIONS AND FUNCTIONS WITH RC-STRONGLY CLOSED GRAPH

The family of regular closed subsets of a topological space is used to introduce two concepts concerning a function f from a space X to a space Y. The first of them is the notion of f being rc-continuous. One of the established results states that a space Y is extremally disconnected if and only if each continuous function from a space X to Y is rc-continuous. The second concept studied is the notion of a function f having an rc-strongly closed graph. Also one of the established results characterizes rc-compact spaces (≡S-closed spaces) in terms of functions that possess rc-strongly closed graph.


Introduction.
A subset A of a topological space (X, T ) is called regular closed if A = Cl(Int(A)), that is, if A is equal to the closure of its interior.It is called semiopen if there exists U ∈ T such that U ⊆ A ⊆ Cl(U).A space (X, T ) is called S-closed (see [10]) if every cover ᐁ of (X, T ) by semiopen sets contains a finite subfamily {U 1 ,...,U n } such that X = n i=1 Cl(U i ).In [1], it is proved that a space (X, T ) is S-closed if and only if every cover of X by regular closed subsets contains a finite subcover for X.This result motivated the study of other covering properties by regular closed subsets; for example, rc-Lindelöf spaces (see [2]), almost rc-Lindelöf spaces (see [3]), rc-paracompact spaces (see [2]), and others.In fact, the name rc-compact space is recently used for Sclosed spaces (see [6]) in order to have a unified terminology when dealing with covering properties by regular closed sets.
On the other hand, in [5], the rc-convergence of a filterbase or a net in a space (X, T ) is formulated using the regular closed sets that contain the limit point.The concept of rc-convergence is proved to be useful in characterizing S-closed (rc-compact) spaces and also extremally disconnected spaces.
In Section 3, we use the family of regular closed subsets to define rc-continuous functions.In our study of this new class of functions, we state a couple of characterizations.Then we show that rc-continuous functions can be used to characterize extremally disconnected spaces.
In Section 4, we introduce, again by using the family of regular closed subsets, the notion of a function f that has an rc-strongly closed graph G(f ).Characterizations of a function to have an rc-strongly closed graph are given and then some results connecting them to rc-compact spaces are stated.In particular, for a Hausdorff space (Y , M), it is shown that (Y , M) is an rc-compact space if and only if certain functions to (Y , M), that have an rc-strongly closed graph, must be rc-continuous.

Definitions and preliminaries.
In what follows, a space always means a topological space with no separation axiom unless it is explicitly stated.For a subset A of a space (X, T ), we let Cl T (A) (or simply Cl(A)) denote the closure of A while Int T (A) (or simply Int(A)) will denote the interior of A, both in the space (X, T ).A subset A of (X, T ) is called semiopen if there exists U ∈ T such that U ⊆ A ⊆ Cl(U ).The family of all semiopen subsets of (X, T ) is denoted by SO(X, T ).A subset A of (X, T ) is called regular closed if A = Cl(Int(A)).We let RC(X, T ) denote the family of all regular closed subsets of (X, T ).It is easy to see that for any space (X, T ), it is true that RC(X, T ) ⊆ SO(X, T ).The complement of a regular closed subset of (X, T ) is called regular open.Equivalently, a subset U is a regular open subset of (X, T ) if U = Int(Cl(U)).We let RO(X, T ) denote the family of all regular open subsets of (X, T ).As a convention, a cover of a space by regular closed subsets will be called an rc-cover.
All definitions and results included in this section are previously known and we quote the reference when possible, while others are easy to establish and we include them for later use.Definition 2.1 (see [10]).A space (X, T ) is called rc-compact (≡ S-closed) if every cover ᐁ of X by semiopen subsets contains a finite subfamily {U 1 ,...,U n } such that X = n i=1 Cl(U i ).
Proposition 2.2 (see [1]).A space (X, T ) is rc-compact if and only if every rc-cover of X contains a finite subcover for X.Definition 2.3.A subset A of a space (X, T ) is called an rc-compact subset of X (≡ S-closed of X in [10]) or an rc-compact subset relative to X (≡ S-closed subset relative to X in [9,10]) if every rc-family Ꮽ of regular closed subsets of X, with A ⊆ Ꮽ, contains a finite subfamily whose union also contains A.
is open for each U ∈ T .In an extremally disconnected space (X, T ), RC(X, T ) = RO(X, T ).Definition 2.5 (see [5,Definition 3.1]).(a) A net (x λ ) λ∈Λ in a space (X, T ) is said to rc-converge to a point x 0 ∈ X if each F ∈ RC(X, T ), with x 0 ∈ F , contains a tail T λ of (x λ ), that is, the net (x λ ) is eventually in each F ∈ RC(X, T ) with x 0 ∈ F .Equivalently, for each U ∈ T such that x 0 ∈ Cl(U), Cl(U) contains a tail of (x λ ).This fact is expressed by writing x λ → (rc)x 0 .
(b) A net (x λ ) λ∈Λ in a space (X, T ) is said to rc-accumulate at a point x 0 ∈ X if for each F ∈ RC(X, T ) and for each λ ∈ Λ, F T λ ≠ ∅, where T λ is the tail of the net (x λ ) determined by λ, that is, (x λ ) is frequently in each F ∈ RC(X, T ) that contains x 0 .Proposition 2.6 (see [5]).A space (X, T ) is rc-compact if and only if every net in (X, T ) rc-accumulates to some point of X. Proposition 2.7 (see [5]).A space (X, T ) is extremally disconnected if and only if each net (x λ ) in X, that is convergent in the usual sense, is also rcconvergent.

rc-continuous functions.
For a space (X, T ) and a point x ∈ X, we let We recall that a function f : In this section, we introduce rc-continuous functions that form a proper subclass of the class of weakly continuous functions.In studying this new class, we state several characterizations of rc-continuous functions and then we show that a space is extremally disconnected if and only if each continuous function onto that space is rc-continuous.
The next result is a direct consequence of the definitions.

Proposition 3.2. Every rc-continuous function is weakly continuous.
However, the converse of this result is not true.To see that, we recall that every continuous function is weakly continuous.On the other hand, it is easy to provide an example of a continuous function that is not rc-continuous.For instance, consider the function f from the set of real numbers R with the usual topology onto itself given by f (x) = x +1.Though this function is continuous, it is not rc-continuous.
The first characterization of rc-continuous functions is obtained through introducing a new topology on the codomain.For a space (X, T ), we introduce the regular closed subsets generated topology, abbreviated TRC, to be the topology on X generated by the subbase RC(X, T ).This topology is interesting in its own and will be studied separately elsewhere.

Proposition 3.3. A function f : (X, T ) → (Y , M) is rc-continuous if and only if the function
Let (X, T ) be a given space and let A ⊆ X.We say that a point x is in the rc-closure of A (rc-Cl(A)) if A F ≠ ∅ for each F ∈ RC(X,T ,x).We say that A is rc-closed if A = rc-Cl(A).We state the following characterizations of rccontinuous functions.
Proposition 3.4.The following conditions are equivalent for a function Next, we show that rc-continuity is characterized by rc-convergence of nets.
. By rc-continuity of f at x 0 , there exists U ∈ ON(x) such that f (U) ⊆ W .But x λ → x 0 , so there exists λ 0 ∈ Λ such that x λ ∈ U for each λ ≥ λ 0 .It follows that f (x λ ) ∈ W for each λ ≥ λ 0 , that is, W contains a tail of f (x λ ).Thus, f (x λ ) → (rc)f (x 0 ).Sufficiency.Suppose that f is not rc-continuous at x 0 .Then there exists Then (D, ≤) is a directed set and we define a net ϕ : D → X as follows: for d ∈ D, say d = (U, t), we let ϕ(d) = x d = t.By its construction, we have that x d → x 0 in X.However, the net (f (x d )) does not rc-converge to f (x 0 ), a contradiction.
We have noted earlier that a continuous function need not be, in general, rccontinuous.However, it is true if the codomain of the function is assumed to be extremally disconnected.In fact, it characterizes extremally disconnected spaces as it is established in our last result of this section.

Theorem 3.6. A space (Y , M) is extremally disconnected if and only if each continuous function from a space (X, T ) into
is open.By the continuity of f and since f (x) ∈ Cl M (V ), there exists U ∈ T such that x ∈ U and f (U) ⊆ Cl M (V ).This proves that f is rc-continuous at the arbitrary point x ∈ X.

)). Consider the identity function i Y : (Y , M) → (Y , M).
Although i Y is continuous, it is not rc-continuous at x 0 , which contradicts the hypothesis.This completes the proof.

Functions with rc-strongly closed graph. The graph of a function
is a closed subset of the product space X ×Y .On the other hand, f is said to have a strongly closed graph (see [4]) if whenever (x, y) ∈ X ×Y and (x, y) ∉ G(f ), there exist U ∈ T and V ∈ M such that x ∈ U , y ∈ V , and In [4], functions that possess strongly closed graph are related to H-spaces.In this section, we define the concept of a function f to have an rc-strongly closed graph, study its properties, and relate them to rc-compact spaces.
By the definitions, it directly follows that if a function f possesses a strongly closed graph, then it possesses an rc-strongly closed graph.
Our first characterization of functions that have rc-strongly closed graph is in terms of rc-convergence.

Proof
Necessity.Assume that f has an rc-strongly closed graph.Let (x λ ) λ∈Λ be a net in X which converges to a point x 0 ∈ X and assume that the net (f (x λ )) rc-converges to a point y 0 ∈ Y .We show that f (x 0 ) = y 0 .Suppose, to the contrary, that (x 0 ,y 0 ) ∉ G(f ).Since G(f ) is rc-strongly closed, then there exist Sufficiency.Suppose that G(f ) is not rc-strongly closed.Then there exists a point (x 0 ,y 0 ) ∈ X ×Y , where (x 0 ,y 0 ) ∉ G(f ), such that whenever U ∈ T and We make D into a directed set in the obvious way and define a net as follows: By its construction, it is clear that the net (x d ) converges in X to x 0 while the net f ((x d )) rc-converges in Y to y 0 .However, f (x 0 ) ≠ y 0 , which contradicts our hypothesis.
The graph of a continuous function need not be rc-strongly closed as it is shown in the next example.
Example 4.4.We take X = R with the usual topology T and Y = R with the left-ray topology M. It is clear that the identity function i X : (X, T ) → (X, M) is continuous.However, the graph G(i X ) is not rc-strongly closed.For instance, (0, 1) ∉ G(i X ) but for any U ∈ T and V ∈ M such that 0 ∈ U and 1 Shortly, we provide a sufficient condition on the codomain of a continuous function to insure that it has an rc-strongly closed graph.But first we need to introduce the following class of spaces.Definition 4.5.A space (X, T ) is called an rc-T 1 -space if for every pair of points x, y ∈ X, with x ≠ y, there exists a regular closed subset F of X such that x ∈ F but y ∉ F .
We remark here that the class of rc-T 1 -spaces coincides with the class of weakly Hausdorff spaces.Recall that a space (X, T ) is called weakly Hausdorff (see [9]) if each singleton {x} is the intersection of regular closed subsets.However, our terminology agrees with the recent trend of naming concepts formulated using regular closed subsets, as it is noted in the introduction.
The following are two easily established facts.Proposition 4.6.Every Hausdorff space is an rc-T 1 -space.Also, every rc-T 1 -space is a T 1 -space.Proposition 4.7.Every extremally disconnected rc-T 1 -space is Hausdorff.
We are now ready to give a sufficient condition for a continuous function f to have an rc-strongly closed graph.Theorem 4.8.Let f be a continuous function from a space (X, T ) to an rc-T 1 -space (Y , M).Then f has an rc-strongly closed graph.
. By continuity of f at x, there exists U ∈ ON(x) such that f (U) ⊆ Y −Cl M (V ).This implies that f (U) Cl M (V ) = ∅ and it follows that (x, y) ∈ U × Cl M (V ) with (U × Cl M (V )) G(f ) = ∅.Thus, G(f ) is rc-strongly closed.
The remaining results of this section relate functions with rc-strongly closed graph to rc-compact spaces.Theorem 4.9.Let (Y , M) be an rc-compact space.If a function f : (X, T ) → (Y , M) has an rc-strongly closed graph, then f is rc-continuous.
Next, we consider the case when x = t.This means that f (x) = y 0 ≠ y.Again, since (Y , M) is a Hausdorff space, there exists V ∈ M such that y ∈ V and y 0 ∉ Cl M (V ).Moreover, the net ϕ : Λ → Y does not rc-accumulate to y.So, there exists W ∈ M and λ ∈ Λ such that y ∈ Cl M (W ) and f (T λ ) Cl M (W ) = ∅ for a tail T λ of Λ.Take F = Cl M (V Cl M (W )).Then y ∈ F , y 0 ∉ F , and F is regular closed as it is easy to see that F = Cl M (V Cl M (W )) = Cl M (V W ).Moreover, f (T λ ) F = ∅.Thus, (T λ {t} × F) G(f ) = ∅.This completes the proof that G(f ) is rc-strongly closed.
Finally, by appealing to Proposition 3.5, we show that f is not rc-continuous.We point here to the fact that the net (λ) λ converges in (X, T ) to the point t.However, the net (λ) λ∈Λ = (f (λ)) λ ∈Λ does not rc-accumulate to y 0 , which means that it is not rc-convergent to y 0 = f (t).Thus, f is not rc-continuous.The established situation contradicts our hypothesis.Thus, (Y , M) is rccompact.
and we can find H ∈ RC(Y , M) such that f (x) ∈ H and H D = ∅.By rc-continuity of f , we find U ∈ ON(x) such that f (U) ⊆ H.It follows that f (U) D = ∅ and so x ∈ U ⊆ X − f −1 (D).This shows that f −1 (D) is closed.(b)⇒(c) follows easily by the fact that if G is regular open then G is rc-closed.(c)⇒(d) is clear.(d)⇒(a) follows by direct application of the definition.