An exploration of semi-open sets in topological spaces

Open sets are the basic structures of topological spaces. Properties of open sets and their relationship to closed sets are well understood. Several topological constructs are defined using open and closed sets, including continuous functions and the separation axioms. Modifications of the definition of open sets have been a topic of interest for many years. In 1963 Norman Levine [3] defined a set A in a topological space to be semi-open provided there exists an open set O such that O ⊆ A ⊆ O. With this definition, we will develop properties of semi-open sets, as well as how they relate to the well-understood notions of open and closed sets. We will explore a definition of semi-closed set, and the relationship between semi-open and semi-closed sets. We also modify the definition of continuity as well as the separation axioms in the context of semi-open and semi-closed sets.