Relations between several topologies obtained by the ideal of countable sets

TL;DR

This paper explores the relationships between various topologies generated by different types of open sets and local functions in the context of the ideal of countable sets, aiming to establish new general relations and conditions for their equality.

Contribution

It introduces new relations between topologies derived from $ heta$-open, $ heta_ ext{omega}$-open, and local functions associated with the ideal of countable sets, including conditions for their equality.

Findings

New relations between topologies are established.

Conditions for the equality of $ ext{omega}$-open and local function topologies are identified.

General results hold in the context of the ideal of countable sets.

Abstract

The main goal of this paper is to investigate relations between topologies obtained by: $\theta$-open sets, $\omega$-open sets, $\theta_\omega$-open sets, local function, and local closure function with ideal of the countable sets. As the result we will obtain some new the relations which hold in general, and, as the ultimate goal, equality of the topology of $\omega$-open sets and topology obtained by the local function with ideal of countable sets.