On Analytical Approach to Semi-open/Semi-closed sets in Metric Spaces”

Abstract

The concept of open and closed sets has been extensively discussed on both metric and topological spaces. Various properties of these sets have been proved under the underlying spaces. However, scanty literature is available about semi-open /semi-closed sets on these spaces. For instance, little effort has been made in introducing these sets as clopen sets in topological spaces but no literature exists of the same under metric spaces. In this paper, with reference to the already existing definitions and properties of open and closed sets in metric spaces as well as in topological spaces we shall present definitions of semi-open/ semiclosed sets and furthermore prove basic properties of these sets on metrics spaces. The results of the study will provide a deeper understanding as well as extension knowledge for the concept of open and closed sets to their somewhat counter-intuitive terms of semi- open /semi-closed.