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.