Various types of continuity and their interpretations in ideal topological spaces

TL;DR

This paper explores various forms of continuity in ideal topological spaces, translating classical concepts into this context and analyzing their relationships and distinctions.

Contribution

It introduces translations of $ heta$-continuity and weak continuity into ideal topological spaces and clarifies their relationships with standard continuity.

Findings

Every $ heta$-continuous function is continuous under certain conditions.

An example of weakly continuous function not being $ au_ heta$-continuous is provided.

The relations between different types of continuity are fully characterized.

Abstract

This paper is a continuation of work started in \cite{njampavcont} on preserving continuity in ideal topological spaces. We will deal with $\theta$-continuity and weak continuity and give their translations in ideal topological spaces. As consequences of those results, we will prove that every $\theta$-continuous function is continuous if topologies are generated by $\theta$-open sets and we will give an example of weakly continuous function which is not $\tau_\theta$-continuous. This will complete the diagram of relations between continuous, $\tau_\theta$-continuous, $\theta$-continuous, weakly continuous and faintly continuous functions.