On $\theta$-extension of continuous mapping

TL;DR

This paper investigates conditions under which a continuous function defined on a dense subset of a topological space can be extended to a weakly-continuous function on the entire space, without assuming separation or compactness.

Contribution

It provides necessary and sufficient conditions for extending continuous mappings to weakly-continuous mappings in general topological spaces.

Findings

Established a theorem characterizing extension existence

Removed the need for separation axioms

Removed the need for compactness assumptions

Abstract

We consider the problem of constructing a weakly-continuous mapping extending continuous mapping defined on a dense set of a topological space to the entire space. Theorem on necessary and sufficient conditions for the existence of such an extension are proved. The axioms of separation and compactness of spaces are not assumed.