  In this paper, some features of countably $\alpha$-compact topological spaces
are presented and proven. The connection between countably $\alpha$% -compact,
Tychonoff, and $\alpha$-Hausdorff spaces is explained. The space is countably
$\alpha$-compact space iff every locally finite family of non-empty subsets of
such space is finite is demonstrated. The countably $% \alpha$-compact space
with weight greater than or equal to $\aleph_0$ is the $\alpha$-continuous
image of a closed subspace of the cube $D^{\aleph_0}$ is discussed. The
boundedness of $\alpha$-continuous functions mapping $\alpha$% -compact spaces
to other spaces is cleared. Moreover, the $\alpha$% -continuous function
mapping the space $X$ to the countably $\alpha$-compact space $Y$ is an
$\alpha$-closed subset of $X\times Y$ is argued and proved. We explained that
the $\alpha$-continuous functions mapping any topological space to a countably
$\alpha$-compact space can be extended over its domain under some constraints.
We claimed that the property of being $\alpha$% -compact is countably
$\alpha$-compact but the converse is not and the countable union of countably
$\alpha$-compact subspaces of $X$ is also countably $\alpha$-compact.
