Embeds in a topological W-group, or a W-space that embeds in a topological group?

Question

In Theorem 3.11 of Tkachuk - A compact space $K$ is Corson compact if and only if $C_p(K)$ has a dense lc-scattered subspace it's shown that if a compact Hausdorff space embeds in a topological W-group (a topological group that's a W-space), then it's a Corson compact.

Is it sufficient to assume that if we have a compact Hausdorff W-space that embeds in a topological group, then we still have a Corson compact?

---

In a direct communication from Tkachuk, he pointed out to me that "embeds in a topological group" is not an interesting topological property, as every (well, certainly he intended to qualify, completely regular) space embeds in some topological group. So the answer to my question is no, but since I could not find a source for this fact on this forum, an answer that shows this in detail might be of value.

Answer 1

No, every Tychonoff space $X$ (so in particular every compact Hausdorff space) embeds (as a closed subspace) in $F(X)$, the free topological group over $X$.

See chapter 7 of Topological groups and related structures by Arhangel’skii and Tkachenko for the details of the construction.

The point is that $F(X)$ is algebraically the free group over $X$, equipped with the finest group topology that makes the inclusion of $X$ a topological embedding. But this is not an honest definition, because the bulk of the work is showing that there is such a topology to begin with.

Answer 2

Or, you can embed $X$ into a Tychonoff cube $[0,1]^\kappa$, where $\kappa$ is the weight of $X$, say. The Tychonoff cube embeds in the corresponding power of the unit circle, say by embedding $[0,1]$ into the circle via $t\mapsto\mathrm{e}^{\mathrm{i}t}$.

Alternatively, the Tychonoff cube is a subspace of the topological vector space $\mathbb{R}^\kappa$.