SOMEWHAT r-CONTINUOUS AND SOMEWHAT r-OPEN FUNCTIONS

In this paper new classes of functions namely somewhat r-continuous and somewhat r-open functions are introduced and studied by making use of regular open sets and regular closed sets. Relationship between the new classes and other classes of functions like somewhat continuous, completely continuous, almost completely continuous etc., are established besides giving examples, counter examples, properties and characterisations. AMS Subject Classification: 54C08, 54C10, 54D10, 54D15


Introduction
Gentre and Hoyle [4] studied the concept of somewhat continuous functions and somewhat open functions.In this paper a new type of somewhat continuous and open functions namely somewhat r-continuous and somewhat r-open func-tions is introduced.These types of functions were discussed in [2] and [3].In section 2 basic definitions are introduced.In section 3 we define somewhat r-continuous functions and study its properties.Characterizations of somewhat r-continuous functions are given and its relation with some other types of functions is also studied.In section 4, r-equivalent topologies for somewhat r-continuity is discussed.Section 5 deals with definition of somewhat r-open function and its properties.

Preliminary Notes
In this article, by (X, τ ), we mean a topological space X with a topology τ on it.Similarly (Y, σ), denotes a topological space Y with a topology σ on it.We abbreviate (X, τ ) and (Y, σ) as X and Y respectively.For a set A, Cl(A) denotes the closure of A and int(A) denotes its interior.A subset A of X is said to be regular open if A=Int(Cl(A)), regular closed if A=Cl(Int(A)).Definition 1.A function f : X → Y is said to be somewhat continuous [4] if for U ∈ σ and f −1 (U ) = φ, there exists an open set V in X such that V = φ and V⊂ f −1 (U ).Definition 2. A function f : X → Y is said to be Cl-supercontinuous [8] (≡ Clopen continuous [6]) if for each x ∈ X and each open set V containing f (x) there exists clopen set U containing x such that f (U ) ⊂ V. Definition 3. A function f : X → Y is said to be δ-continuous [6] if for each x∈ X and for each regular open set V containing f (x) there exists a regular open set U containing x such that f (U ) ⊂ V.
3. Somewhat r-Continuous Functions Definition 6.Let X and Y be any two topological spaces.A function f : X → Y is said to be somewhat r-continuous if for U ∈ σ and f −1 (U ) = φ, there exists a regular open set V in X such that V =φ and V⊂ f −1 (U ).

Example 7. Let
Then f is somewhat r-continuous.
Then f is somewhat continuous, but not somewhat r-continuous.Definition 11.A topological space is locally indiscrete [9] if every open set is closed.
Theorem 12. Let f : X → Y be somewhat continuous and X be locally indiscrete.Then f is somewhat r-continuous.Every completely continuous function is somewhat rcontinuous.
Proof.The proof is obvious.
Remark 18.The opposite proposition does not hold.
Let f : X → Y be the identity function.Then f is somewhat r-continuous, but f is not completely continuous.
Theorem 20.If X is a discrete space and f : X → Y is somewhat r-continuous, f is completely continuous.
Definition 32.Let M be a subset of a topological space (X, τ ).Then M is said to be r-dense in X if there is no regular closed set C in X such that M ⊂ C ⊂ X.
(3)⇒(2) Suppose ( 2) is not true.This means that there exists a closed set Theorem 34.Let (X, τ ) and (Y, σ) be any two topological spaces.Let A be a regular open set of X and f : (A, τ /A) → (Y, σ) be somewhat rcontinuous such that f (A) is dense in Y. Then any extension F of f is somewhat r-continuous.

Case (1): (
This can be proved by using the same argument as in case (1).

Case (3): (
The proof follows from the proofs of case(1) and (2).Definition 36.A topological space X is said to be r-separable if there exists a countable subset B of X which is r-dense in X.
Theorem 38.If f is somewhat r-continuous function from X onto Y and if X is r-separable, Y is separable.
Proof.Let f: X → Y be somewhat r-continuous function such that X is r-separable.Then there exists a countable set B of X which is r-dense in X.Then f (B) is dense in Y. Since B is countable and f is onto, f (B) is countable.So Y is separable.

r-Weakly Equivalent Topologies
Definition 39.If X is a set and τ and σ are topologies for X then τ is said to be weakly equivalent [4] to σ provided if U∈ τ and U = φ, then there is an open set V in (X, σ) such that V = φ and V⊂ U and if U in σ and U = φ then there is an open set V in (X, τ ) such that V = φ and V ⊂ U.
Definition 40.If X is a set and τ and σ are topologies for X then τ is said to be r-weakly equivalent to σ provided if U∈ τ and U = φ, then there is a regular open set V in (X, σ) such that V = φ and V⊂ U and if U in σ and U = φ then there is a regular open set V in (X, τ ) such that V = φ and V ⊂ U.

Proof. Let U be any open set in (
Then f is somewhat continuous, but not somewhat r-continuous.
Definition 52.A topological space is locally indiscrete [9] if every open set is closed.
Theorem 53.Let f : X → Y be somewhat continuous and X be locally indiscrete.Then f is somewhat r-continuous.
This can be proved by using the same argument as in case (1).

Proof.
The proof follows from the result, clopen sets are regular open.Theorem 13.Every Cl-supercontinuous function is somewhat r-continuous.Proof.The proof follows from the result, clopen sets are regular open.Remark 14.The opposite proposition does not hold.Example 15.Let X = {a, b, c, d}, Y = {a, b, c}, τ = {X, φ, {a, b}, {a}{b}, {a, b, d}}, σ = {Y, φ, {a}, {b}}.Let f : X → Y be the identity function.Then f is somewhat r-continuous, but f is not Cl-super continuous.Theorem 16.Let f : X → Y be somewhat r-continuous where X is locally indiscrete.Then f is Cl-super continuous.Proof.The proof follows from the result that regular open set is open and open set in a locally indiscrete space is clopen.Theorem 17.

Proof.
Proof follows from the result that finite union of regular open sets in a discrete space is regular open.Corollary 21.If X is finite, T 1 and f : X → Y is somewhat r-continuous, f is completely continuous.Theorem 22. Every somewhat r-continuous function is δ-continuous.Proof.Let f : X → Y be somewhat r-continuous.Let V be non empty regular open set in Y. Then it is open.Since f is somewhat r-continuous there exists a regular open set U such that U⊂ f −1 (V ).So f is δ-continuous.Remark 23.The opposite proposition does not hold.Example 24.Let X = Y = {a, b, c}, τ = {X, φ, {a, b}, {c}}, σ = {Y, φ, {a}, {b}, {b, c}}.
and X is a discrete space, then f is almost completely continuous.Proof.Proof follows from the result that finite union of regular open sets in a discrete space is regular open.Corollary 28.If f : X → Y is somewhat r-continuous and X is finite and T 1 , then f is almost completely continuous.Proof.Proof follows from the result that finite union of regular open sets in a finite T 1 space is regular open.Remark 29.The following diagram shows the relationship between somewhat continuous, somewhat r-continuous, completely continuous, closure super continuous and δ-continuous functions.

Theorem 35 .
Let (X, τ ) and (Y, σ) be any two topological spaces.X=A∪B where A and B are regular open subsets of X.Let f : (X, τ ) → (Y, σ) be a function such that f /A and f /B are somewhat r-continuous.Then f is a somewhat r-continuous function.Proof.Let U be any open set in (Y, σ) such that f −1 (U ) = φ.Then either (f /A) −1 (U ) = φ or (f /B) −1 (U ) = φ or both (f /A) −1 (U ) and (f /B) −1 (U ) = φ.

Theorem 67 .
If f : X → Y is almost completely continuous and Y is locally indiscrete, f is somewhat r-continuous.Proof.Let V be open in Y. Since Y is locally indiscrete, V is clopen and so regular open.Since f is almost completely continuous f −1 (V ) = U is regular open.That is there exists a regular open set U such that U ⊂ f −1 (V ) for each open set V in Y. Therefore f is somewhat r-continuous.Theorem 68.If f : X → Y is somewhat r-continuous and X is a discrete space, then f is almost completely continuous.Proof.Proof follows from the result that finite union of regular open sets in a discrete space is regular open.Corollary 69.If f : X → Y is somewhat r-continuous and X is finite and T 1 , then f is almost completely continuous.Proof.Proof follows from the result that finite union of regular open sets in a finite T 1 space is regular open.Remark 70.The following diagram shows the relationship between somewhat continuous, somewhat r-continuous, completely continuous, closure super continuous and δ-continuous functions.Somewhatcontinuous cl − supercontinuousտ ↓ Completelycontinuous → Somewhatr−continuous ↓ ↓ Almostcompletelycontinuous → δ − continuous that f −1 (C) ⊂ D. This means f −1 (C) is r-dense in X.But by (3)f (f −1 (C))=C must be dense in Y, a contradiction to choice of C. So (2) is true.(2)⇒(1) Let U ∈ σ and f −1 (U ) = φ.Then Y-U is closed in Y and f −1 (Y − U )=X-f −1 (U ) = X.So by (2) there exists a proper regular closed subset D of X such thatD ⊃ f −1 (Y − U )=X-f −1 (U ).That is X-D ⊂ f −1 (U )and X-D is a non empty regular open subset.So f is somewhat r-continuous.Theorem 75.Let (X, τ ) and (Y, σ) be any two topological spaces.Let A be a regular open set of X and f :

Theorem 76 .
Let (X, τ ) and (Y, σ) be any two topological spaces.X=A∪B where A and B are regular open subsets of X.Let f : (X, τ ) → (Y, σ) be a function such that f /A and f /B are somewhat r-continuous.Then f is a somewhat r-continuous function.Proof.Let U be any open set in (Y, σ) such that f −1 Proof.Let U be open set in (Y, σ * ) such that f −1 (U ) = φ.Then U = φ.Since σ and σ * are weakly equivalent there exists an open set W in (Y, σ) such that W = φ and W ⊂ U. Then f −1 (W ) = φ.Since f : (X, τ ) → (Y, σ) is somewhat r-continuous there exists regular open set