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Following Gruenhage we call a topological space $X$ cometrizable if $X$ admits a weaker metrizable topology such that every point $x\in X$ has a (not necessarily open) neighborhood base consisting of metrically closed sets.

It is easy to see that each cometrizable space is regular. A typical example of a cometrizable space is the Sorgenfrey line, i.e., the real line endowed with the topology, generated by the base $\{[a,b):a<b\}$.

Problem. Is each cometrizable Tychonoff space a subspace of a cometrizable topological group?

A particular case of this problem is especially interesting.

Question. Is the Sorgenfrey line homeomorphic to a subspace of a cometrizable topological group?

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During the night sleep my brain has found affirmative answers to both problems. The answer to the Problem is rather long, so I will present only the answer to Question, which is a bit tricky. First a definition.

A subset $D$ of a topological space $X$ is called $k$-dense in $X$ if each compact subset $K\subset X$ can be enlarged to a compact set $\tilde K$ such that $D\cap\tilde K$ is dense in $\tilde K$.

A topological space $X$ is called $k$-separable if it contains a countable $k$-dense subset.

It is easy to see that each dense subset of a metrizable space is $k$-dense. So, metrizable separable spaces are $k$-separable.

Also each dense subset of the Sorgenfrey line $\mathbb S$ is $k$-dense, so $\mathbb S$ is $k$-separable, too.

Theorem. For any $k$-separable space $X$ and cometrizable space $Y$, the function space $C_k(X,Y)$ with compact-open topology is cometrizable.

Proof. Let $\tau$ be the weaker metrizable topology on $Y$ witnessing that $Y$ is cometrizable. Denote by $Y_\tau$ the metrizable topological space $(Y,\tau)$. Let $D$ be a countable $k$-dense subset of $X$. Consider the restriction operator $R:C_k(X,Y)\to Y_\tau^D$, $R:f\mapsto f{\restriction}D$. Let $\sigma$ be the metrizable topology on $C_k(X,Y)$ such that the map $R:(C_k(X,Y),\sigma)\to Y^D_\tau$ is a topological embedding. It can be shown that the topology $\sigma$ witnesses that the function space $C_k(X,Y)$ is cometrizable. $\square$

Since the Sorgenfrey line $\mathbb S$ is $k$-separable, Theorem implies

Corollary. The function space $C_k(\mathbb S,\mathbb R)$ is cometrizable.

Now it remains to observe that the Sorgenfrey line $\mathbb S$ admits a topological embedding to its own function space $C_k(\mathbb S,\mathbb R)$ (which is a cometrizable Abelian topological group).

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    $\begingroup$ Why downvote (for both problem and solution)? Was the problem trivial or the soltion evident? I have written the problem after thinking three days and realizing that my former arguments (using free Abelian groups) do not work. But this trick with k-separability finally did the job. So, what is wrong? $\endgroup$ – Taras Banakh Jan 25 at 8:52

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