New Kinds of Open Sets

In this paper, we introduced new types of open sets which we called p*open set and semi p*open set. Besides, we get the following results: Every open set is p*open and semi p*open. Examples are given to show that the converse may not be true . Preopen set and p*open set are equivalent. Every semi p*open set is semi p open set.

A covering property with respect to generalized preopen sets

In this paper, we introduce and study the notion of $\mu$-precompact spaces on the observation that each $\mu$-preopen set of a generalized topological space is contained in a $\mu$-open set. The $\mu$-precompactness is weaker than $\mu$-compactness but stronger than weakly $\mu$-compactness of generalized topological spaces.

Characterizations of Strongly Compact Spaces

A topological space(X,τ)is said to be strongly compact if every preopen cover of(X,τ)admits a finite subcover. In this paper, we introduce a new class of sets called -preopen sets which is weaker than both open sets and -open sets. Where a subsetAis said to be -preopen if for eachx∈Athere exists a preopen setUxcontainingxsuch thatUx−Ais a finite set. We investigate some properties of the sets. Moreover, we obtain new characterizations and preserving theorems of strongly compact spaces.

On certain types of notions via preopen sets

In this paper, we deal with the new class of pre-regular $p$-open sets in which the notion of preopen set is involved. We characterize these sets and study some of their fundamental properties. We also present two other notions called extremally $p$-discreteness and locally $p$-indiscreteness by utilizing the notions of preopen and preclosed sets by which we obtain some equivalence relations for pre-regular $p$-open sets. Moreover, we define the notion of regular $p$-open sets by utilizing the notion of pre-regular $p$-open sets. We investigate some of the main properties of these sets and study their relations to pre-regular $p$-open sets.