# Is every uncountable, homogeneous connected $T_2$-space isomorphic to a subspace of $\mathbb{R}^\omega$?

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 $$\mathbb{R}^\omega$$?

• How about the weak topology on a separable Banach space? It's not metrizable nor even first countable, so can't be homeomorphic to any subspace of $\mathbb{R}^\omega$. – Nate Eldredge Nov 5 '18 at 15:17

According to this research in $$\pi$$-base, the sigma product of incountably many copies of $$\mathbb{R}$$ and the boolean product topology on $$\mathbb{R}^{\omega}$$ are connected, $$T_2$$ and homogeneous.
However, they are not first countable, hence they cannot be homeomorphic to subspaces of $$\mathbb{R}^{\omega}$$.
• Well, there is probably a mistake in $\pi$-base, since the second space I gave is the box topology in $\mathbb{R}^{\omega}$, which is not connected (Steen-Seebach, Counterexamples in Topology, p. 128-129). – Francesco Polizzi Nov 5 '18 at 15:54
• If you actually click on the "connected" listing for that space, it says "this is wrong, please delete me". It seems that the Github code for $\pi$-base correctly says "false": github.com/pi-base/data/blob/master/spaces/S000107/properties/…. I don't know why they are out of sync. The "this is wrong" text doesn't even appear in the git history of that file. – Nate Eldredge Nov 5 '18 at 22:51
Or take the long line: $$\omega_1\times[0,1)$$ with the lexicographic order, minus the first point. Every bounded open interval is isomorphic to $$(0,1)$$, so it is homogeneous. It is first-countable but not second-countable, hence not embeddable into $$\mathbb{R}^\omega$$.