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On the other hand, we have many positive results.

On the other hand, we have many positive results, see this preprint for more information.

A Tychonoff space $X$ is Ascoli if the canoncal 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.

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 $$f:X\to\prod_{k\in\omega}\ell_1(\mathcal N_k)^\omega,\;\;f:x\mapsto \big((f_{k,n}(x))_{n\in\omega}\big)_{n\in\omega}.$$

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.

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 $$f:X\to\prod_{k\in\omega}\ell_1(\mathcal N_k)^\omega,\;\;f:x\mapsto \big((f_{k,n}(x))_{n\in\omega}\big)_{k\in\omega}.$$

On the other hand, we have the following positive result.

Theorem 1. Each stratifiable cosmic space $X$ is cometrizable.

Being cosmic, the space $X$ has a countable network $\mathcal N$ consisting of closed subsets of $X$. Since the cosmic space $X$ is perfectly normal, for every $N\in\mathcal N$ and $n\in\omega$, there exists a continuous function $f_{N,n}:X\to [0,1]$ such that $f_{N,n}^{-1}(0)=N$ and $f_{N,n}^{-1}(1)=X\setminus W_n[N]$. Consider the continuous function $$f:X\to[0,1]^{\mathcal N\times\omega},\;\;f:x\mapsto (f_{N,n}(x))_{(N,n)\in\mathcal N\times\omega},$$and observe that it is injective. Let $\tau$ be the (metrizable) topology on $X$ such that the map $f:(X,\tau)\to [0,1]^{\mathcal N\times \omega}$ is a topological embedding. It follows that for any $N\in\mathcal N$ and $n\in\omega$ the set $N$ is $\tau$-closed and the set $W_n[N]$ 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 normality 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 $\mathcal N'=\{N\in\mathcal N:N\subset X\setminus\overline{U}\}$. Observe that the union $W=\bigcup_{N\in\mathcal N'}W_n[N]$ is a $\tau$-open set such that $$X\setminus\overline{U}\subset W=\bigcup_{N\in\mathcal N'}W_n[N]\subset W_n[X\setminus\overline{U}]\subset W_n[F]\subset X\setminus V_x.$$ 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 $X\setminus W\subset O_x$. $\square$

In fact, Theorem 1 is a partial case of the following more general theorem that can be proved by the same method.

Theorem 2. Each stratifiable space is cometrizable.

On the other hand, we have many positive results.

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.

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 canoncal 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

Corollary 2. Each Ascoli $\aleph_0$-space is cometrizable. In particular, each sequential $\aleph_0$-space is cometrizable.

Finally, we have

Theorem 2. Each stratifiable space $X$ is cometrizable.

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 $$f:X\to\prod_{k\in\omega}\ell_1(\mathcal N_k)^\omega,\;\;f:x\mapsto \big((f_{k,n}(x))_{n\in\omega}\big)_{n\in\omega}.$$

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 $X\setminus W\subset O_x$. $\square$