RARELY s∗g-CONTINUOUS FUNCTIONS

The notion of rare continuity was introduced by Popa. In the present paper, we introduce a new class of functions, called rarely sg-continuous functions and investigate some of its fundamental properties. AMS Subject Classification: 54B05, 54C08, 54D05


Introduction
In 1961, Levine introduced the concept weak continuity [7] in topological spaces.As a generalization of this concept, Popa [12] introduced the notion of rare continuity in the year 1979 and it has been further investigated by Long and Herrington [9] in 1982.Jafari [5,6] studied some more properties of them.Moreover, Popa [10,11] studied the notions of Weakly continuous multifunctions and rarely continuous multifunctions.Meanwhile, Levine [8] introduced the concept of generalized closed sets of a topological space as a generalization of closed sets to the larger family and a class of topological spaces called T 1 2 -spaces.Dunham [3], Dunham and Levine [4] and Caldas [1] further studied some properties of generalized closed sets and T 1 2 -spaces.Caldas and Jafari [2] introduced the concept of rare g-continuity in topological spaces as a generalization of rare continuity and weak continuity.They investigated several properties of rarely g-continuous functions.The notion of I.g-continuity is also introduced which is weaker than g-continuity and stronger than rare g-continuity.Further they showed that when the codomain of a function is regular, then the notions of rare g-continuity and I.g-continuity are equivalent.
In the present paper, we introduce a new class of functions called rarely s * g-continuous functions and investigate some of its fundamental properties.

Preliminaries
Throughout this paper, X and Y are topological spaces.The closure of a set A is the intersection of all closed sets that contain A and is denoted by cl(A) and the interior of a set A is the union of all open sets contained in A and is denoted by int(A).Now we shall require some known definitions.
The family of all open {resp.g-open, s * g-open} sets will be denoted by O(X) {resp.GO(X), S * GO(X)} and the family of all open {resp.g-open, s * g-open} sets containing the point x ∈ X will be denoted by O(X, x) {resp.GO(X, x), S * GO(X, x)}.(f) weakly g-continuous if for each x ∈ X and each open set G containing f (x), there exists U ∈ GO(X, x) such that f (U ) ⊆ Cl(G).
(g) rarely continuous if for each x ∈ X and each Let X = Y = {a, b, c}, τ = {φ, X, {a}, {b, c}}, σ = {φ, Y, {b}, {a, c}} and f : (X, τ ) → (Y, σ) be a identity function.Then f is a rarely s * g-continuous Theorem 3.3.The following statements are equivalent for a function f : X → Y : (1) The function f is rarely s * g-continuous at x ∈ X. ( Then by (3), there exists U ∈ S * GO(X, x) We define the following notion which is a new generalization of s * g-continuity.
If f has this property at each point x ∈ X, then we say that f is I.s * gcontinuous on X. Theorem 3.5.Let Y be a regular space.Then the function f : X → Y is I.s * g-continuous on X if and only if f is rarely s * g-continuous on X.
Proof.We prove only the sufficient condition since the necessity condition is evident.Let f be rarely s * g-continuous on X and x ∈ X. Suppose that We say that a function f : and thus f is weakly s * g-continuous.
Theorem 3.7.If f : X → Y is rarely s * g-continuous function, then the graph function g : X → X × Y , defined by g(x) = (x, f (x)) for every x ∈ X is rarely s * g-continuous.
Proof.Suppose that x ∈ X and W is any open set containing g(x).It follows that there exist open sets U and V in X and Y , respectively, such that Therefore, g is rarely s * g-continuous.
Definition 3.8.Let A = {G i } be a class of subsets of X.By rarely union sets of A we mean {Gi ∪ R G i }, where each R G i is a rare set such that each of Recall that, a subset B of X is said to be rarely almost compact relative to X if every open cover of B by open sets of X, there exists a finite subfamily whose rarely union sets cover B.
A topological space X is said to be rarely almost compact if the set X is rarely almost compact relative to X.A topological space X is called S * GO-compact if every cover of X by s * g-open sets has a finite subcover.Theorem 3.9.Let f : X → Y be rarely s * g-continuous and K a S * GOcompact set relative to X. Then f (K) is rarely almost compact subset relative to Y .

Proof. Suppose that Ω is an open cover of f (K). Let B be the set of all
Hence there is a finite subfamily {U k } k∈∆ which covers K, where ∆ is a finite subset of K.The subfamily Recall that a space X is called T S -space if every s * g-closed set in X is closed in X. Theorem 3.12.Let f : X → Y be a rarely s * g-continuous and X be a door space.Then f is rarely continuous.
Proof.It is an immediate consequence of Lemma 3.11 and Theorem 3.10.Lemma 3.13.(Long and Herrington [9]).If g : Y → Z is continuous and one-to-one, then g preserves rare sets.Theorem 3.14.If f : X → Y is rarely s * g-continuous and g : Y → Z is continuous and one-to-one, then gof : X → Z is rarely s * g-continuous.
Proof.Suppose that x ∈ X and (gof )(x) ∈ V , where V is an open set in V .By hypothesis, g is continuous, therefore there exists an open set G ⊆ Y containing f (x) such that g(G) ⊆ V .Since f is rarely s * g-continuous, there exist a rare set Theorem 3.15.Let f : X → Y be a pre-s * g-open surjection and g : Y → Z a function such that gof : X → Z is rarely s * g-continuous.Then g is rarely s * g-continuous.

Upper (Lower) Rarely s * g-Continuous Multifunctions
We provide the following definitions which will be used in the sequel.Let F : X → Y be a multifunction.The upper and lower inverses of a set V ⊆ Y are denoted by F + (V ) and . Then, by (iii) there exists U ∈ S * GO(X, x) Theorem 4.4.The following are equivalent for a multifunction F : X → Y : . Then P is a rare set and P ∩cl(V ) = φ.Moreover, we have x ∈ s * g-int[F \ (V ∪ R V )] ⊆ s * g-int{F \ [P ∪ cl(V )]}.
(iii) ⇒ (iv): Let V be any regular open set of Y such that F (x) ∩ V = φ.By (iii), there exists a rare set R V with cl(V ) ∩ R V = φ such that x ∈ s * g-int{F \ [cl(V ) ∪ R V ]}.Put P = R V ∪ [cl(V ) \ V ], then P is a rare set and V ∩ cl(P ) = φ.Moreover, we have x ∈ s * g-int{F (iv) ⇒ (i): Let V ∈ O(Y ) such that F (x)∩V = φ.Then F (x)∩int[cl(V )] = φ and int[cl(V )] is regular open in Y .By (iv), there exists a rare set R V with V ∩ cl(R V ) = φ such that x ∈ s * g-int[F − (V ∪ R V )].Therefore, there exists U ∈ S * GO(X, x) such that U ⊆ F − (V ∪ R V ).Hence F (u) ∩ (V ∪ R V ) = φ for each u ∈ U .This shows that F is lower rarely s * g-continuous at x.
e) weakly continuous if for each x ∈ X and each open set G containing f (x), there exists U ∈ O(X, x) such that f (U ) ⊆ Cl(G).

Theorem 3 . 10 .
Let f : X → Y be rarely s * g-continuous and X a T Sspace.Then f is rarely continuous.A space X is called a door space if every subset of X is either open or closed.W. Dunham [[3] Corollary 3.7] proved the following result: Lemma 3.11.A door space is a T 1 2 -space.
/lower rarely s * g-continuous if it is upper/lower rarely s * g-continuous at each point of X.Definition 4.2.A multifunction F : X → Y is said to be (i) upper weakly s * g-continuous at x ∈ X if for each V ∈ O(Y, F (x)), there exist U ∈ S * GO(X, x) such that F (U ) ⊆ cl(V ), (ii) lower weakly s * g-continuous at x ∈ X if for each V ∈ O(Y ) with F (x) ∩ V = φ, there exists U ∈ S * GO(X, x) such that F (u) ∩ cl(V ) = φ for every u ∈ U ,(iii) upper/lower weakly s * g-continuous if it is upper/lower weakly s * g-continuous at each point of X.