On Countably $\alpha$-Compact Topological Spaces
Eman Almuhur, Muhammad Ahsan Khan

TL;DR
This paper explores properties of countably lpha-compact topological spaces, their relationships with other classes, and the behavior of lpha-continuous functions, providing new characterizations and extension results.
Contribution
It introduces new characterizations of countably lpha-compact spaces, examines their relationship with lpha-Hausdorff and Tychonoff spaces, and studies properties of lpha-continuous functions.
Findings
Countably lpha-compact spaces are characterized by finite locally finite families.
Such spaces are lpha-continuous images of closed subspaces of the cube D^{\u001alpha_0}.
lpha-continuous functions from these spaces are lpha-closed subsets of product spaces.
Abstract
In this paper, some features of countably -compact topological spaces are presented and proven. The connection between countably % -compact, Tychonoff, and -Hausdorff spaces is explained. The space is countably -compact space iff every locally finite family of non-empty subsets of such space is finite is demonstrated. The countably -compact space with weight greater than or equal to is the -continuous image of a closed subspace of the cube is discussed. The boundedness of -continuous functions mapping % -compact spaces to other spaces is cleared. Moreover, the % -continuous function mapping the space to the countably -compact space is an -closed subset of is argued and proved. We explained that the -continuous functions mapping any topological…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Computability, Logic, AI Algorithms
