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 $\mathbb{R}^\omega$?



Is every uncountable, homogeneous connected $T_2$-space isomorphic to a subspace of $\mathbb{R}^\omega$?
According to [this search](https://topology.pi-base.org/spaces?q=%7EFirst%20Countable%20%2B%20%24T_2%24%20%2B%20Connected%20%2B%20Homogeneous) in $\pi$-base, the *sigma product of incountably many copies of* $\mathbb{R}$ (no longer in pi-Base as of 2023 March) and the [*boolean product topology on* $\mathbb{R}^{\omega}$](https://topology.pi-base.org/spaces/S000107) are connected, $T\_2$ and homogeneous.


However, they are *not* first countable, hence they cannot be homeomorphic to subspaces of $\mathbb{R}^{\omega}$.


