On Countably $α$-Compact Topological Spaces

Abstract

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 $\aleph_0$ is the $α$-continuous image of a closed subspace of the cube $D^{\aleph_0}$ 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 $X\times Y$ 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.