The rapid development, in the few short years since their introduction, of the closely related condensedand pyknotic mathematics frameworks of Clausen and Scholze, and Barwick and Haine, respectively, forms the immediate background for this talk. In the standard heuristic, these are frameworks for ``doing algebra with objects carrying a topology''; more concretely (but still imprecisely), they are frameworks for coordinated approaches to a wide variety of mathematical objects via their (pre)sheaf representations over the category of profinite sets (i.e., of totally disconnected compact Hausdorff spaces). In our talk's first half, we will introduce and survey these frameworks. In its second half, we will discuss some of the questions in infinitary combinatorics and set theory which these frameworks pose (with a particular focus on the ``condensed image'' of the Whitehead Problem), as well as some answers. This work is joint with Chris Lambie-Hanson.