Separability of subspaces of homogeneous topological spaces 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.
 A: This fails even for topological groups.  For example, the separable topological group $\mathbb{Z}^{\mathfrak{c}}$ contains a subgroup of uncountable cellularity (hence certainly not separable!).  This is a result of my colleague Vladimir Uspenskij in the paper below:
Uspenskij, V. V., On the Suslin number of subgroups of products of countable groups, Acta Univ. Carol., Math. Phys. 36, No. 2, 85-87 (1995). ZBL0854.20064. 
A: Any compact homogeneous hereditarily separable space has size at most $\mathfrak{c}$ (this is a result of Ismail). Thus any compact homogeneous separable space of bigger size provides a counterexample (e.g. $2^\mathfrak{c}$ as in Anonymous' comment).
A: There are Banach spaces $X$ for which the dual space $X^*$ is separable in the weak* topology yet the unit ball $B_{X^*}$ of $X^*$ is not. A notable example is $X=J\!L_2$, the Johnson–Lindenstrauss space.

W.B. Johnson, and J. Lindenstrauss, Some remarks on weakly compactly generated Banach spaces, Israel J. Math. 17 (1974), 219–230. Correction ibid 32
  No. 4 (1979), 382–383.

