SEMI GENERALIZED OPEN SETS AND GENERALIZED SEMI CLOSED SETS IN TOPOLOGICAL SPACES

A BSTRACT . In this paper we introduce a new class of semi generalized open sets, generalized semi closed sets in topological spaces


INTRODUCTION
The study of generalized closed sets in topological space was initiated by Levine in [23].
introduced and investigated semi closed, -open and -closed, pre-open, semi*-open, sg-closed, gsclosed, gp-closed, g-closed, g* closed, s*g-closed, w-closed, g*-closed respectively. Topology is an important and interesting area of mathematics, the study of which will not only introduce to new concepts and theorems but also put into context old ones like continuous functions [1]. However, to say just this is to understate the significance of topology. It is so fundamental that its influence is evident in almost every other branch of mathematics [3]. Topological notions like compactness, connectedness and denseness areas basic to mathematicians of today as sets and functions were to those of last century. Topology has several different branches, genera l topology, algebraic topology, differential topology and topological algebra, the first, general topology, being the door to the study of the others [3,5].
The aim of this paper is to introduce the concept of semi generalized open Sets and generalized semi closed Sets in topological spaces, and provide Semi generalized open Sets and generalized semi closed Sets in topological spaces.

PRELIMINARIES
The union of any (finite or infinite) number of sets in τ belong to τ, The intersection of any two sets in τ belongs to τ.
The pair ( , ) is called a topological space.

Definition 2.2
Let be a non -empty sets and let τ be the collection of all subsets of . Then τ is called discrete topology on the set . The topological space ( , ) is called a discrete space.

OPEN SETS AND CLOSED SETS IN
Hence 1 ∪ 2 ∪ … ∪ is a closed set, as required. So (iii) is true.
The proof of (ii) is similar to that of (iii).

Example 3.5.
On any set X there is the trivial topology {∅, X}. There is also the discrete topology whereas any subset of X is open. Thus, on a set there can be many topologies.

SEMI GENERALIZED * b CLOSED SETS
In this part, we introduce semi generalized* b -closed set and investigate some of its properties.  To see that B ⊂ Y ∩ A ̅ , note that ̅ is closed in , so Y ∩ A ̅ is closed in and contains .
Hence it contains the closure of A in .
To prove the opposite inclusion, note that is closed in , hence has the form B = Y ∩ C for some C that is closed in . Then A ⊂ B ⊂ C, so C is closed in and contains . Hence A ̅ ⊂ C and Y ∩ A ̅ ⊂ Y ∩ C = B.

CONCLUSION
The aim of this paper is to introduce the concepts of semi generalized open Sets and generalized semi closed Sets in topological spaces sets and we study some of their properties.
Furthermore, we discuss the conditions which are added to these concepts in order to coincide with the concept of semi-closed.