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.