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?


At the moment I know the answer to the Question (but not to the Problem). 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).

| cite | improve this answer | |
  • 3
    $\begingroup$ 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 '19 at 8:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.