2022/05/23 by Eman Almuhur, Almuhur, Eman, Muhammad Ahsan Khan +1
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN)
paper · pdf · doi:10.48550/arxiv.2205.11674
openalex publication_date 2022/05/23 · openalex created_date 2022/05/27 · openalex updated_date 2026/07/28
In this paper, some features of countably α-compact topological spaces are presented and proven. The connection between countably α% -compact, Tychonoff, and α-Hausdorff spaces is explained. The space is countably α-compact space iff every locally finite family of non-empty subsets of such space is finite is demonstrated. The countably % α-compact space with weight greater than or equal to ℵ0 is the α-continuous image of a closed subspace of the cube Dℵ0 is discussed. The boundedness of α-continuous functions mapping α% -compact spaces to other spaces is cleared. Moreover, the α% -continuous function mapping the space X to the countably α-compact space Y is an α-closed subset of X× Y is argued and proved. We explained that the α-continuous functions mapping any topological space to a countably α-compact space can be extended over its domain under some constraints. We claimed that the property of being α% -compact is countably α-compact but the converse is not and the countable union of countably α-compact subspaces of X is also countably α-compact.