New classes of compact-type spaces

TL;DR

This paper introduces three new classes of compact-type spaces inspired by classical concepts like $ ext{k}$-spaces and sequential spaces, explores their properties, relationships, and applications in topological groups.

Contribution

It defines new classes of compact-type spaces, characterizes their properties, and relates them to existing classes, including applications to topological groups and locally compact abelian groups.

Findings

New classes of spaces characterized by boundary point convergence.

Relationships established with classical topological classes.

Applications to topological groups and locally compact abelian groups.

Abstract

Being motivated by the notions of $\kappa$-Fr\'{e}chet--Urysohn spaces and $k'$-spaces introduced by Arhangel'skii, the notion of sequential spaces and the study of Ascoli spaces, we introduce three new classes of compact-type spaces. They are defined by the possibility to attain each or some of boundary points $x$ of an open set $U$ by a sequence in $U$ converging to $x$ or by a relatively compact subset $A\subseteq U$ such that $x\in \overline{A}$. Relationships of the introduced classes with the classical classes (as, for example, the classes of $\kappa$-Fr\'{e}chet--Urysohn spaces, (sequentially) Ascoli spaces, $k_{\mathbb R}$-spaces, $s_{\mathbb R}$-spaces etc.) are given. We characterize these new classes of spaces and study them with respect to taking products, subspaces and quotients. In particular, we give new characterizations of $\kappa$-Fr\'{e}chet--Urysohn spaces and show that each feathered topological group is $\kappa$-Fr\'{e}chet--Urysohn. We describe locally compact abelian groups which endowed with the Bohr topology belong to one of the aforementioned classes. Numerous examples are given.