New classes of compact-type spaces
Saak Gabriyelyan, Evgenii Reznichenko
TL;DR
The paper introduces three new compact-type space classes—open-compact attainable, weakly open-compact attainable, and $\kappa$-sequential spaces—motivated by $\kappa$-Fréchet--Urysohn and Ascoli notions, and investigates their relationships with classical classes such as Fréchet--Urysohn, $k$-spaces, and $k_{\mathbb{R}}$-spaces. It develops a framework based on $\kappa$-pseudo open maps and weakly $\kappa$-pseudo open maps to characterize these spaces, proves permanence properties under quotients and certain images, and analyzes stability under products, subspaces, and quotients. The work includes a suite of examples showing the non-reversibility of implications and delineates notable cases such as feathered topological groups and locally compact abelian groups with the Bohr topology, yielding new characterizations of $\kappa$-Fréchet--Urysohn spaces and their relation to Ascoli-type behavior. Overall, the paper broadens the understanding of compact-type spaces and their behavior under standard topological constructions, with implications for function spaces and topological groups.
Abstract
Being motivated by the notions of $κ$-Fréchet--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 $κ$-Fréchet--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 $κ$-Fréchet--Urysohn spaces and show that each feathered topological group is $κ$-Fréchet--Urysohn. We describe locally compact abelian groups which endowed with the Bohr topology belong to one of the aforementioned classes. Numerous examples are given.