Could not include topos theory - contents
CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A closed cover of a topological space $X$ is a collection $\{U_i \subset X\}$ of closed subsets of $X$ whose union equals $X$: $\cup_i U_i = X$. Usually it is also required that every point $x \in X$ is in the interior of one of the $U_i$.
Closed covers can be obtained from open covers by forming the closure of each of the open subsets. The result clearly satisfies the clause that every point is in the interior of one of the closed subsets.
Applications of closed covers in Čech homology is discussed in
Related discussion is also in this MO thread
In analytic geometry, affinoid domains have closed sets as analytic spectra and hence the topological space underlying a Berkovich analytic spaces is equipped by a closed cover by affioid domains