gs Λ Continuous Function in Topological Space

In 1986, Maki [1] continued the work of Levine and Dunham on generalized closed sets and closure operators by introducing the notion of Λ-sets in topological spaces. A Λ-set is a set A which is equal to its kernel (= saturated set), that is, to the intersection of all open supersets of A. Arenas et al. [2] introduced and investigated the notion of λ-closed sets and λ-open sets by involving Λ-sets. In 2008Caldas et al. [3] introduced Λ generalized closed sets (Λg, Λ-g, gΛ) and their properties. They also studied the concept of λ closedmaps. In 2007, Caldas et al. [4] introduced the concept of λ irresolute maps. In this paper, we establish a new class of maps called gsΛ continuous function and study its properties and characteristics. Throughout this paper, (X, τ), (Y, σ), and (Z, η) (or simply X,Y, and Z will always denote topological spaces on which no separation axioms are assumed unless explicitly stated. Int(A), Cl(A), IntλA, ClλA, gsΛCl(A), and gsΛInt(A) denote the interior of A, closure of A, lambda interior of A, lambda closure of A, gs Lambda closure of A and gs Lambda interior of A, respectively.


Introduction
In 1986, Maki [1] continued the work of Levine and Dunham on generalized closed sets and closure operators by introducing the notion of Λ-sets in topological spaces.A Λ-set is a set  which is equal to its kernel (= saturated set), that is, to the intersection of all open supersets of .Arenas et al. [2] introduced and investigated the notion of -closed sets and -open sets by involving Λ-sets.
In 2008 Caldas et al. [3] introduced Λ generalized closed sets (Λg, Λ-g, gΛ) and their properties.They also studied the concept of  closed maps.In 2007, Caldas et al. [4] introduced the concept of  irresolute maps.
In this paper, we establish a new class of maps called gsΛ continuous function and study its properties and characteristics.
Throughout this paper, (, ), (, ), and (, ) (or simply , , and  will always denote topological spaces on which no separation axioms are assumed unless explicitly stated.Int(), Cl(), Int  , Cl  , gsΛCl(), and gsΛInt() denote the interior of , closure of , lambda interior of , lambda closure of , gs Lambda closure of  and gs Lambda interior of , respectively.
The complement sets of semi open (resp., preopen and regular open) are called semi closed sets (resp., preclosed and regular closed).The semiclosure (resp., preclosure) of a subset  of  denoted by sCl(), (pCl()) is the intersection of all semi closed sets (pre closed sets) containing .
A topological space (, ) is said to be (1) a generalized closed [7] if Cl() ⊂ , whenever  ⊂ and  is open in (, ), (2) a  * closed [8]  (ii) [12] a  ĝ space if every ĝ closed subset of  is closed in , (iii) [12] a   space if every gs closed subset of  is closed in .
(4) In  1 space every gsΛ closed set is  closed.
(6) In partition space every gsΛ closed set is  closed and ĝ closed.(7) In a door space every subset is gsΛ closed.(8) In  1/2 space every subset is gsΛ closed.Definition 6.Let (, ) be the topological space and  ⊆ .We define gsΛ closure of  (briefly gsΛCl()) to be the intersection of all gsΛ closed sets containing , gsΛ interior of  (briefly gsΛInt()) to be the union of all gsΛ open sets contained in .
Lemma 7 (see [10]).Let A and B be subsets of a topological space (, ).The following properties hold.
(2) If A is gsΛ closed then A = gsΛ().Converse need not be true.

Theorem 9. Every continuous function is gsΛ continuous function.
Proof.Let  be a closed set in (, ) and a function  : (, ) → (, ) be a continuous function.Hence,  −1 () is closed in (, ).As every closed set is gsΛ closed set by Proposition 5, we have  −1 () is gsΛ closed in .Thus,  is a gsΛ continuous function.
Converse need not be true as seen from the following example.
Proof.Let  be a closed set in (, ) and a function  : (, ) → (, ) be a  continuous function.Hence,  −1 () is  closed in (, ).As every  closed set is gsΛ closed set by Proposition 5, we have  −1 () is gsΛ closed in X.Thus,  is a gsΛ continuous function.
Converse need not be true as seen from the following example.

Theorem 14. Every ĝ continuous function is gsΛ continuous function.
Proof.As every ĝ closed set is gsΛ closed set [10]  is a gsΛ continuous function.
Converse need not be true as seen from the following example.Proof.The proof is simple as in a partition space every gsΛ closed set is ĝ closed [10].

Theorem 17. Every contra continuous function is gsΛ continuous function.
Proof.Let  be a closed set in (, ) and a function  : (, ) → (, ) be a contra continuous function.Hence, As every open set is gsΛ closed set by Proposition 5, we have  −1 () is gsΛ closed in .Thus,  is a gsΛ continuous function.
Converse need not be true as seen from the following example.
Lemma 25 (see [17]  (iii) For each  in (, ), the inverse of every neighbourhood of () is a gsΛ neighbourhood of .
(iv) For each  in (, ), and each neighbourhood  of (), there is a gsΛ neighbourhood  of , such that () ⊆ .
Proof.(i)⇔(ii).Let  : (, ) → (, ) be a gsΛ continuous function and  be an open set in (, ).Then   is closed set in .Hence, by the definition of gsΛ continuous function The converse is analogous.
This completes the proof of the theorem.

On Composite Functions
Theorem 36.Composition of continuous functions is a gsΛ continuous function.
Proof.Since every closed set is a gsΛ closed set, the proof is simple to prove.
Theorem 37. Composition of  irresolute functions is a gsΛ continuous function.
(ii) The proof is clear as in a  1 space every gsΛ closed set is  closed by Proposition 5.

( 6 )( 7 )
gsΛInt() is the largest gsΛ open set contained in .If  is gsΛ open then  = gsΛint().Converse need not be true.Proofs are obvious from the definition and properties of gsΛ closed sets and gsΛ open sets.

Theorem 31 .
Let  : (, ) → (, ) be a function.Then the following are equivalent.(i) The function is gsΛ continuous.(ii) The inverse of each open set in (, ) is gsΛ open in (, ).
Sufficiency.Assume that for each gsΛ neighbourhood   of  in (, ),  ∩   ̸ = .Suppose  ∉ gsΛCl().Then there exist a gsΛ closed set  of (, Theorem 32.Let {  / ∈ } be any family of topological spaces.If  :  → Π  is a gsΛ continuous function, then    of :  →   is gsΛ continuous for each  ∈ , where    is the projection of Π  on   .Proof.We will consider a fixed  ∈ .Suppose   is an arbitrary open set in   .Then    −1 (  ) is open in Π  .Since  is a gsΛ continuous function, we have  −1 (   −1 (  )) = (   ∘ ) −1 (  ) is gsΛ open in .Hence,    of is a gsΛ continuous function.Definition 33.Let  be a subset of . mapping  : (, ) →  is called a gsΛ continuous retraction if  is gsΛ continuous and the restriction   is the identity mapping on .Definition 34.A topological space (, ) is called a gsΛ Hausdroff if for each pair ,  of distinct points of , there exist gsΛ neighbourhoods  1 and  2 of  and , respectively, that are disjoint.Let  be a subset of  and  :  →  be a gsΛ continuous retraction.If  is gsΛ Hausdroff, then  is a gsΛ closed set of .Proof.Suppose that  is not gsΛ closed.Then there exist a point  in  such that  ∈ gsΛCl() but  ∉ .It follows that () ̸ =  because  is gsΛ continuous retraction.Since  is gsΛ Hausdroff there exist disjoint gsΛ open sets  1 and  2 in  such that  ∈  1 and () ∈  2 .Now let  be an arbitrary gsΛ neighbourhood of .Then  ∩  1 is a gsΛ neighbourhood of .Since  ∈ gsΛCl(), by Theorem 30, we have ( ∩  1 ) ∩  ̸ = 0. Therefore, there exist a point  ∈  ∩  1 ∩ .Since  ∈ , we have () =  ∈  1 and hence () ∉  2 .This implies that () ̸ ⊆  2 because  ∈ .This is a contrary to gsΛ continuity of .Consequently,  is a gsΛ closed set of .