On preserving continuity in ideal topological spaces

TL;DR

This paper investigates conditions under which continuity is preserved when mapping between topological spaces with induced topologies, especially focusing on open and closed functions and their behavior under idealization.

Contribution

It provides new sufficient conditions for continuity preservation in ideal topological spaces, including cases involving open and closed functions, with illustrative examples.

Findings

Conditions for continuity preservation are established.

Open and closed functions are analyzed in the context of ideal topologies.

Examples show the necessity of the proposed conditions.

Abstract

We present some sufficient conditions for continuity of the mapping $f:\langle X,\tau_X^*\rangle \to \langle Y,\tau_Y^*\rangle$, where $\tau_X^*$ and $\tau_Y^*$ are topologies induced by the local function on $X$ and $Y$, resp. under the assumption that the mapping from $\langle X, \tau_X \rangle$ to $\langle Y, \tau_Y \rangle$ is continuous. Further, we consider open and closed functions in this matter, as we state the cases in which the open (or closed) mapping is being preserved through the "idealisation" of both domain and codomain. Through several examples we illustrate that the conditions we considered can not be weakened.