Application of αδ -Closed Sets

In this paper, we introduce the notion of αδ - US spaces. Also we study the concepts of αδ -convergence, sequentially αδ -compactness, sequentially αδ -continunity and sequentially αδ - sub-continuity and derive some of their properties.


Introduction
In 1967, A. Wilansky [1] introduced and studied the concept of spaces.Also, the notion of αδ-closed sets of a topological space is discussed by R. Devi, V. Kokilavani and P. Basker [2,3].The concept of slightly continuous functions is introduced and investigated by Erdal Ekici et al. [4].In this paper, we define that a sequence n US  x in a space X is αδ-converges to a point x X  if   n x is eventually in every αδ-open set con- taining x .Using this concept, we define the αδ-US space, Sequentially-αδ-continuous, Sequentially-Nearlyαδ-continuous, Sequentially-Sub-αδ-continuous and Sequentially-αδO-compact of a topological space   , X  .

Preliminaries
Throughout this paper, spaces X and Y always mean topological spaces.Let X be a topological space and A, a subset of X.The closure of A and the interior of A are denoted by and

Int
, the δ-interior [5] of a subset A of X is the union of all regular open sets of X contained in A and is denoted by Alternatively, a set , where . The family of all δ-open (resp.δ-closed) sets in X is denoted by (resp. ).A subset and the complement of a α-open are called α-closed.The intersection of all α-closed sets containing A is called the α-closure of A and is denoted by , Dually, α-interior of A is defined to be the union of all α-open sets contained in A and is denoted by We recall the following definition used in sequel.Definition 2.1.A subset A of a space X is said to be (a) An α-generalized closed [7]


The intersection of all αδ-closed sets of X containing A is called αδ-closure of A and is denoted by
Theorem 3.9.The product space of an arbitrary family of αδ-US topological space is an αδ-US topological space.

Proof. Let  
:  x  be a family of αδ-US topological spaces with the index set Δ.The product space of

Sequentially αδO-Compact Preserving Functions
 , then it is said to be sequentially αδ-continuous.
(b) Sequentially-Nearly-αδ-continuous, if for each se- f are sequentially αδ-continuous functions, the sequence x and hence x E  .This shows that is sequentially αδclosed.
Proof.Let : f X Y  be a sequentially nearly αδcontinuous function and M be any sequentially αδO- compact subset of X .We will show that    .Since f is sequentially nearly αδ-continuous, there exists a subsequence is sequentially sub-αδ-continuous.
Sufficiency: Let M be any sequentially αδO-compact set of X .We will show that  

Slightly αδ-Continuous
is said to be slightly αδ-continuous if for each x X  and for each is the family of clopen sets containing f(x) in a space .
Suppose that f is not slightly αδ-continuous at a point x X  , then there exists a Slightly αδ-continuous.Theorem 5.5.The following are equivalent for a function : f X Y  : (a) f is slightly αδ-continuous, (b) for each x X  and for each Proof.Suppose that is clopen 1 .For any distinct points T  .Proof.For any pair of distinct points x and in y X , there exist disjoint clopen sets U and in such that e., a set is δ-open if it is the union of regular open sets.The complement of a δ-open set is called δ-closed.
(c) Sequentially-Sub-αδ-continuous if for each point x X  and each sequence   n x in αδ-converging to, there exists a subsequence  

Y Theorem 4 . 5 .
implies that   f M is a sequentially αδO-compact set of .Every sequentially αδO-compact preserving function is sequentially sub-αδ-continuous.Proof.Suppose that : f X Y  is a sequentially O  -compact preserving function.Let x be any point of X and   n x a sequence that αδ-converges to x .We denote the set   converges to x , M is sequentially αδO-compact.By hypothesis, f is sequentially αδO-compact subset of .Now in Y that f sequentially sub-αδ-continuous.Theorem 4.6.A function : f X  Y is sequentially O  -compact preserving if and only if


f M .Then for each n N  , there exists a point .Since   n x is a sequence in the sequentially αδO-compact set M there exists a subsequence  k n x of   n x that αδ-converges to a point in M .By hypothesis to y  f(M).This implies that f(M) is sequentially αδO-compact in .tially sub-αδ-continuous and f M is sequentially αδ-closed in for each sequentially αδO-compact set M of Y X , then f is sequentially αδO-compact preserv- ing.Proof.It will be sufficient to show that -αδ-continuous for each sequentially αδO-compact set M of X and by Lemma 3.3.We have already done.So, let   n x be any sequence in M that αδ-converges to a point x M  .Then, since f is sequentially sub-αδ-continuous there exists a subsequence   in the sequentially αδ- which is a contradiction.Hence   A space is called αδ-regular if for each αδ-closed set F and each point x F Definition 5.9.A space is said to be αδ-normal if for every pair of disjoint αδ-closed subsets 1 and f(V) are disjoint open sets.This shows that is clopen regular.If is slightly αδ-continuous injective open function from a αδ-normal space B are disjoint open sets.Thus, Y is cl- open normal. f