# On some functional generalizations of the regularity of topological spaces

## TL;DR

This paper explores generalized notions of regularity in topological spaces, linking these concepts to continuity properties and characterizing regularity in first-countable Hausdorff spaces via the absence of a specific topological structure.

## Contribution

It introduces new generalizations of regular spaces motivated by continuity considerations and characterizes regularity in first-countable Hausdorff spaces through the Gutik hedgehog.

## Key findings

- First-countable Hausdorff space is regular iff it lacks a Gutik hedgehog
- New generalizations of regularity are proposed based on continuity properties
- Characterization of regularity via topological substructures

## Abstract

We introduce and study some generalizations of regular spaces, which were motivated by studying continuity properties of functions between (regular) topological spaces. In particular, we prove that a first-countable Hausdorff topological space is regular if and only if it does not contain a topological copy of the Gutik hedgehog.