F−open and F−closed sets in Topological Spaces

Abstract

An open (resp., closed) subset $A$ of a topological space $(X,\mathcal{T})$ is called {\it $F$-open} (resp., $F$-closed) set if $cl(A)\setminus A$ (resp., $A\setminus int(A)$) is finite set. In this work, we study the main properties of these definitions and examine the relationships between $F$-open and $F$-closed sets with other kinds such as regularly open, regularly closed, closed, and open sets. Then, we establish some operators such as $F$-interior, $F$-closure, and $F$-derived...etc., using $F$-open and $F$-closed sets. At the end of this work, we introduce definitions of $F$-continuous function, $F$-compact space, and other related properties.