The situation is this. I have a space $X$ which is second countable, compact, and Hausdorff (it's a modified form of a type space, though I don't think that matters here). It has size continuum. It may or may not have isolated points.


Must $X$ have a discrete subset of size continuum?


The obvious inductive constructions yield only countable discrete sets, and there isn't obviously a Zorn's Lemma argument, since the ascending union of discrete sets is not necessarily discrete. But this is not my specialty; is this known one way or the other?


The purpose of this is to produce a many-models theorem, and seemingly has nothing to do with topology.



Must uncountable compact Hausdorff spaces have large discrete subsets?
Without even using compactness (or metrisability), a subspace of a second-countable space is second-countable. There is no discrete, uncountable, second-countable space.


