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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 $\mathbb{R}^\omega$?

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    $\begingroup$ 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 $\mathbb{R}^\omega$. $\endgroup$ – Nate Eldredge Nov 5 '18 at 15:17
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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 $\mathbb{R}^{\omega}$.

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    $\begingroup$ 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). $\endgroup$ – Francesco Polizzi Nov 5 '18 at 15:54
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    $\begingroup$ 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. $\endgroup$ – Nate Eldredge Nov 5 '18 at 22:51
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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 $\mathbb{R}^\omega$.

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