TITLE: Must uncountable compact Hausdorff spaces have large discrete subsets?
QUESTION [5 upvotes]: 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.

REPLY [11 votes]: Without even using compactness (or metrisability), a subspace of a second-countable space is second-countable. There is no discrete, uncountable, second-countable space.