Many commonly-studied properties of topological spaces relate to, either by definition or result, one of the following:

i) the existence of a subcollection or a refinement with some property of interest for each open cover of the space X,

ii) the existence of a sequence of open covers of X with some property of interest, or

iii) the existence of a selection (or refinement of such) with some property of interest for each sequence of open covers of the space X.

We'll discuss several such properties that typical students in the graduate topology sequence may not have seen, results and applications related to them, and a few open questions.