Here is another example, but with a somewhat different flavor than the others posted here. Namely, let Q be the set of rational numbers, and consider the collection of all functions from Q to Q which can be extended to continuous functions from R to R. (Here R is the set of real numbers, endowed with the Euclidean topology.) The article "Can a Subset's Topology Detect Continuous Extensions?" (The College Mathematics Journal, Volume 49, 2018 - Issue 2) contains a proof that there is no topology T for which this collection equals the collection of all continuous self-maps of Q with respect to T.
Mike Krebs
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