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In this paper, are investigated some new weak separation axioms connected with "Gem set". Besides are both introduced and studied some of their properties and the relation between them and other weak separation axioms is highlighted .

This paper introduced a new compactness space called Gem-compact space , pointcompact space and Prefect-compact space under the idea of "Gem-Set" in topological spaces and study some of their properties and relations among them with basic compact space.

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In this section, we will explain some concepts such as ( C -local function, C -closure operator, C -local function, C * -topology and compatible of center ideal space) with clarification of relationships.