A regular topological space $X$ is called


$\bullet$ *cosmic* if $X$ is a continuous image of a separable metrizable space;


$\bullet$ *[cometrizable](http://www.ams.org/journals/tran/1989-313-01/S0002-9947-1989-0992600-5/S0002-9947-1989-0992600-5.pdf)* if $X$ admits a weaker metrizable topology such that each point $x\in X$ has a (not necessarily open) neighborhood base consisting of metrically closed sets.


In 1989 [Gruenhage](http://www.ams.org/journals/tran/1989-313-01/S0002-9947-1989-0992600-5/S0002-9947-1989-0992600-5.pdf) proved that under PFA a cometrizable space is cosmic if and only if it contains no uncountable discrete subspace and no uncountable subspace of the Sorgenfrey line.


Can the cometrizability be moved to the right-hand part of this characterziation?



> 
> **Question.** Is each cosmic space cometrizable? Maybe under PFA or OCA?
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Is each cosmic space cometrizable?
Discussing this problem with [Alex Ravsky](https://mathoverflow.net/users/43954/alex-ravsky) we constructed the following



> 
> **Example.** The Euclidean topology $\tau\_0$ on the set $\mathbb Q$ of rational numbers can be enlarged to a regular topology $\tau$ of weight $\omega\_1$ such that the countable (and hence cosmic) topological space $(\mathbb Q,\tau)$ is not cometrizable.
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The topology $\tau$ is constructed by transfinite induction of length $\omega\_1$. At each step $\alpha<\omega\_1$ we define a regular second countable topology $\tau\_\alpha$ on $\mathbb Q$ such $\bigcup\_{\beta<\alpha}\tau\_\beta\subset\tau\_\alpha\subsetneq \tau\_{\alpha+1}$ that the space $(\mathbb Q,\tau\_\alpha)$ has no isolated points and for every $\alpha<\omega\_1$ the topology $\tau\_{\alpha+1}$ contains a neighborhood $U\_\alpha$ of zero such that for every $\beta\ge\alpha$ and any neighborhood $V\in\tau\_\beta$ of zero the $\tau\_\alpha$-closure of $V$ is not contained in $U\_\alpha$. Then the topology $\tau=\bigcup\_{\alpha\in\omega\_1}\tau\_\alpha$ is a desired regular topology for which the space $(\mathbb Q,\tau)$ is not cometrizable. $\square$


On the other hand, we have many positive results, see [this preprint](https://arxiv.org/abs/1901.10987) for more information. 


For topological spaces $X,Y$ by $C\_k(X,Y)$ we denote the space of continuous functions from $X$ to $Y$, endowed with the compact-open topology.



> 
> **Theorem 1.** For any $\aleph\_0$-space $X$ and any cometrizable space $Y$ the function space $C\_k(X,Y)$ is cometrizable.
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*Proof.* It is well-known that the $\aleph\_0$-spaces are images of metrizable separable spaces under compact-covering maps, which implies that $C\_k(X,Y)$ embeds into the function space $C\_k(M,Y)$ over some metrizable separable space $M$. So, we can assume that the space $X$ is metrizable and separable. Let $D$ be a countable dense set in $X$. Let $\tau$ be a metrizable topology witnessing that the space $Y$ is cometrizable. It can be shown that the topology on $C\_k(X,Y)$ inherited from the Tychonoff power $(Y,\tau)^D$ witnesses that the function space $C\_k(X,Y)$ is cometrizable.


**Corollary 1.** For any $\aleph\_0$-space $X$ the function spaces $C\_k(X)=C\_k(X,Y)$ and $C\_k(C\_k(X))$ are cometrizable.


A Tychonoff space $X$ is *Ascoli* if the canonical map $\delta:X\to C\_k(C\_k(X))$ assigning to each $x\in X$ the Dirac measure $\delta\_x:f\mapsto f(x)$ is a topological embedding. It is known that each $k$-space is Ascoli.


The definition of an Ascoli space and Corollary 1 implies 



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> **Corollary 2.** Each Ascoli $\aleph\_0$-space is cometrizable. In particular, each sequential $\aleph\_0$-space is cometrizable.
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Finally, we have 



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> **Theorem 2.** Each stratifiable space $X$ is cometrizable.
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*Proof.* Since $X$ is stratifiable, every point $x\in X$ has a decreasing system of open neighborhoods $(W\_n(x))\_{n\in\omega}$ such that each closed set $F\subset X$ is equal to $\bigcap\_{n\in\omega}\overline{W\_n[F]}$ where $W\_n[F]=\bigcup\_{x\in F}W\_n(x)$. 


It is known that each stratifiable space is a $\sigma$-space. So $X$ has a $\sigma$-discrete network $\mathcal N$, which can be written as the countable union $\mathcal N=\bigcup\_{k\in\omega}\mathcal N\_k$ of discrete families in $X$.
Since stratifiable spaces are paracompact, each set $N\in\mathcal N\_k$ has an open neighborhood $O\_N\subset X$ such that the family $(O\_N)\_{N\in\mathcal N\_k}$ is discrete in $X$. 


For every $k,n\in\omega$ and $N\in\mathcal N\_k$ consider the open neighborhood 
$$O\_{k,N,n}=O\_N\cap W\_n[N]$$ of $N$. Since stratifiable spaces are perfectly normal, there exists a continuous function $f\_{k,N,n}:X\to [0,1]$ such that $f\_{k,N,n}^{-1}(1)=N$ and $f\_{k,N,n}^{-1}(0)=X\setminus O\_{k,N,n}$.


Let $\ell\_1(\mathcal N\_k)$ be the Banach space of all functions $h:\mathcal N\_k\to\mathbb R$ such that $\|h\|:=\sum\_{N\in\mathcal N\_k}|h(N)|<+\infty$.
For every $N\in\mathcal N\_k$ let $e\_N:\mathcal N\_k\to\{0,1\}$ be the unique function such that $e\_N^{-1}(1)=\{N\}$. So $(e\_N)\_{N\in\mathcal N\_k}$ is the standard unit basis of the Banach space $\ell\_1(\mathcal N\_k)$.


Consider the continuous function $$f\_{k,n}:X\to\ell\_1(\mathcal N\_k),\;\;f\_{k,n}:x\mapsto\sum\_{N\in\mathcal N\_k}f\_{k,N,n}(x)e\_N.$$
Next, consider the continuous (injective) function
$$f:X\to\prod\_{k\in\omega}\ell\_1(\mathcal N\_k)^\omega,\;\;f:x\mapsto \big((f\_{k,n}(x))\_{n\in\omega}\big)\_{k\in\omega}.$$


Let $\tau$ be the metrizable topology on $X$ such that the function $f:(X,\tau)\to\prod\_{k\in\omega}\ell\_1(\mathcal N\_k)^\omega$ is a topological embedding.
It follows that for any $k,n\in\omega$ and $N\in\mathcal N\_k$ the set $N$ is $\tau$-closed and the set $O\_{k,N,n}$ is $\tau$-open.


We claim that the topology $\tau$ witnesses that the space $X$ is cometrizable. 
Given any point $x\in X$ and an open neighborhood $O\_x\subset X$ of $x$, use the regularity of $X$ to find an open neighborhood $U$ of $x$ such that $\overline{U}\subset O\_x$. Consider the closed set $F=X\setminus U$ and observe that $F=\bigcap\_{n\in\omega}\overline{W\_n[F]}$. Since $x\notin F$, there exists $n\in\omega$ such that $x\notin\overline{W\_n[F]}$. Then $V\_x:=X\setminus\overline{W\_n[F]}$ is an open neighborhood of $x$.


Since $\mathcal N$ is a network, the open set $X\setminus\overline{U}\subset F$ coincides with the union $\bigcup\mathcal N'$ of the subfamily $\mathcal N'=\{N\in\mathcal N:N\subset X\setminus\overline{U}\}$. For every $k\in\omega$ let $\mathcal N\_k'=\mathcal N\_k\cap\mathcal N'$. Observe that the union $W=\bigcup\_{k\in\omega}\bigcup\_{N\in\mathcal N'\_k}O\_{k,N,n}$ is a $\tau$-open set such that $$X\setminus\overline{U}\subset W=\bigcup\_{k\in\omega}\bigcup\_{N\in\mathcal N'\_k}O\_{k,N,n}\subset W\_n[X\setminus\overline{U}]\subset W\_n[F]\subset X\setminus V\_x.$$
Then $V\_x\subset X\setminus W\subset \overline{U}\subset O\_x$ and the $\tau$-closure of the neighborhood $V\_x$ is contained in $X\setminus W\subset O\_x$. $\square$


