In this paper, we introduce and study a new class of sets namely -closed sets which settled in between the class of closed sets and the class of g-closed sets and then we study many basic properties of -closed set together with the relationships of some other sets. As applications of -closed sets, we introduce some new separation properties, namely -spaces, -spaces and -spaces and then discuss some of their properties. Further we introduce and study new types of continuity namely -continuity and -irresoluteness.
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