We say a space $(X,\tau)$ is *homogeneous* if for any $x,y\in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$.

What is an example of a connected, homogeneous $T_2$-space $(X,\tau)$ with $|X| = 2^{\aleph_0}$ such that $(X,\tau)$ is not homeomorphic to a subspace of $\mathbb{R}^\omega$?