C-open Sets on Topological Spaces

TL;DR

This paper introduces and explores C-open and C-closed sets in topological spaces, analyzing their properties, related operators, and implications for continuous maps and compactness.

Contribution

It presents the novel concepts of C-open and C-closed sets, investigates their properties, and applies them to continuous maps and compact spaces.

Findings

C-open and C-closed sets have distinct properties from classical open and closed sets.

Operators like interior, closure, and frontier are redefined using C-sets.

Continuous maps and compactness are characterized via C-open and C-closed sets.

Abstract

An open (resp., closed) subset A of a topological space (X, T ) is called C-open (resp., C-closed) set if cl(A) \ A (resp., A \ int(A)) is a countable set. This paper aims to present the concept of C-open and C-closed sets. We first investigate their basic properties. Then, we found some operators such as interior, closure, limit, border, and frontier using C-open and C-closed sets. The relationships between them are clarified and discussed. Finally, we exhibit continuous maps and compact space defined using C-open and C-closed sets and scrutinize their main properties.