On RC-spaces

TL;DR

This paper investigates RC-spaces, a class of regular T_1-spaces with a specific base property, showing that certain well-known spaces like Niemytzki and Sorgenfrey planes are RC-spaces.

Contribution

It introduces RC-spaces, characterizes their properties, and identifies specific examples such as Niemytzki and Sorgenfrey planes as RC-spaces.

Findings

Normal spaces are RC-spaces

Niemytzki plane is an RC-space

Sorgenfrey plane is an RC-space

Abstract

Following Frink's characterization of completely regular spaces, we say that a regular T_1-space is an RC-space whenever the family of all regular open sets constitutes a regular normal base. Normal spaces are RC-spaces and there exist completely regular spaces which are not RC-spaces. So the question arises, which of the known examples of completely regular and not normal spaces are RC-spaces. We show that the Niemytzki plane and the Sorgenfrey plane are RC-spaces.