Are spaces that can be embedded in a topological W-group first countable?

Question

This question is linked to Are W-spaces with countable pseudocharacter first countable?.

Here I state the following definition:

A topological space $X$ is embeddable in a topological W-group if it is homeomorphic to a subspace of a W-space that has a group topology (see also: W-space and has a group topology).

I am thinking about more possible counterexamples to answer to my previous question (cited at the beginning), and I got interested in knowing whether the Peng-Wu Group or the Weak topology on separable Hilbert space are W-spaces or not.

After this, I noticed that there are no examples of spaces that are embeddable in a topological W-group, but that aren't first countable. So, additionally, I was wondering if anyone knows an example of such, or if embeddable in a topological W-group implies first countable.

Answer 1

An example of a space $X$ that is embeddable in a topological W-group, but is not first countable, is the Alexandroff one-point compactification of a uncountable discrete space, for example Fort space on the real numbers.

It is embeddable in a $\Sigma$-product $Y$ of copies of $\mathbb R$. Specifically, $$Y=\{x=(x_\alpha)\in\mathbb R^\Gamma:\{\alpha:x_\alpha\ne 0\} \text{ is countable}\}$$ for some set $\Gamma$ of uncountable cardinality. The space $Y$ is given the topology of pointwise convergence, induced from the product topology on $\mathbb R^\Gamma$. It is a topological group under addition (subgroup of $(\mathbb R^\Gamma,+)$) and is a $W$-space because any $\Sigma$-product of W-spaces is a W-space.

The space $X$ is homeomorphic to the subspace $\{e_z:z\in\Gamma\}\cup\{\mathbf 0\}\subseteq Y$, where $e_z$ is the characteristic function of $\{z\}$ and $\mathbf 0$ is the zero function. See Is the one-point compactification of an uncountable discrete space Eberlein compact?.

See also π-Base, Search for Embeds in a topological $W$-group + ~First countable.