# Is each cometrizable space a subspace of a cometrizable topological group?

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.

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?

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).

• 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? – Taras Banakh Jan 25 at 8:52