Let $\ X\ $ be a homogenous separable topological space (i.e. for every $\ x\ y\in X\ $ there exists a homeomorphism $\ f:X\to X\ $ such that $\ f(x)=y,\ $ and there is a countable dense subset of $\ X).\ $
Question Is every subspace of $\ X\ $ separable?
The question allows several variations by considering different stronger kinds of homogeneity. The other direction would be to consider different separability properties like Hausdorff, normal, etc. (No, forget the metric case :) ).
I would conjecture that in the simple case of homogeneity there should be an example of $\ X,\ $ and of its subspace $\ Y\ $ which is not separable while $\ X\ $ is.