Is a quotient of a locally compact separable metric space submetrisable? Let $X$ be a quotient of a locally compact separable metric space $Y$. Is $X$ submetrisable (i.e. does $X$ have a coarser topology that is metrisable)?
 A: The space $X$ can fail to be even Hausdorff (for instance, if $Y$ is non-empty, $Z$ is a proper dense subset of $Y$, $x^\star\not\in Y\setminus Z$, $X=(Y\setminus Z)\cup \{ x^\star \}$, and $q:Y\to X$ be the quotient map such that $q(x)=x$ for each $x\in Y\setminus Z$ and $q(x)=x^\star $ for each $x\in Z$.
On the other hand, we have the following
Proposition. Let $X$ be a $T_1$ and $T_3$ space, which is a continuous image of a space with a countable network. Then $X$ is submetrizable.
Proof. Since the space $X$ has a countable network, the space $X^2$ is hereditarily Lindelöf.
We claim that the diagonal $\Delta=\{(x,x)\in X^2\}$ is a $G_\delta$ subset of $X^2$. Indeed, let $(x,y)\in X^2\setminus\Delta$ be any point. Since $X^2$ is Hausdorff there exist open in $X$ disjoint neighborhoods $U_{(x,y)}$ and  $V_{(x,y)}$ of the points $x$ and $y$, respectively. Since the space $X$ is $T_3$, there exist open in $X$ disjoint neighborhoods $U’_{(x,y)}$ and $V’_{(x,y)}$ of the points $x$ and $y$, respectively such that $\overline{U’_{(x,y)}}\subset U_{(x,y)}$ and $\overline{V’_{(x,y)}}\subset V_{(x,y)}$. A family $$\{ U’_{(x,y)}\times V’_{(x,y)}:(x,y)\in X^2\setminus\Delta\}$$ is an open cover of the space $X^2\setminus \Delta$. Since the latter space is Lindelöf, there exists a countable subset $Z$ of $X^2\setminus\Delta$ such that a family $\{ U’_{z}\times V’_{z}:z\in Z\}$ is a cover of $X^2\setminus\Delta$. Then $\Delta=\bigcup_{z\in Z} X^2\setminus \overline{U’_{z}}\times \overline{V’_{z}}$.
Since the space $X$ is Lindelöf, by Theorem 3.8.11 from [Eng], it is paracompact. By Corollary 2.9 from [Gru] (see also [Bor] and [Oku]), $X$ is submetrizable.
Theorem 3.8.11. [Eng] Every open cover of a Lindelöf space has a locally finite open refinement.
Proof. Let $\mathcal U$ be an open cover of a Lindelöf space $X$. The space $X$ being regular, for every $x\in X$ there exist open sets $U_x,V_x\subset X$ such that $x\in U_x\subset\overline{U_x}\subset V_x$ and $V_x$ is contained in a member of $\mathcal U$. Let $\{U_{x_i}\}_{i=1}^\infty$ be a countable subcover of the cover $\{U_x\}_x\in X$ of
the space $X$. The sets $$W_i = V_{x_i}\setminus \left(U_{x_1}\cup U_{x_2}\cup\dots U_{x_{i-1}}\right),\mbox{ where }i = 1,2,\dots,$$ are open and constitute a cover of $X$. Indeed, for any $x\in X$ we have $x\in W_{i(x)}$ where $i(x)$ is the smallest integer $i$ satisfying $x\in V_{x_i}$. The cover $\{W_i\}_{i=1}^\infty$ is a refinement of $\mathcal U$ and is locally finite because $U_{x_j}\cap W_i=\varnothing$ for $i> j$. $\square$
References
[Bor] C.R. Borges, On stratifiable spaces, Pacific J. Math., 17,  1–16.
[Eng]  Ryszard Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.
[Gru] Gary Gruenhage Generalized Metric Spaces, in: K.Kunen, J.E.Vaughan (eds.) Handbook of Set-theoretic
Topology, Elsevier Science Publishers B.V., 1984.
[Oku] A. Okuyama, On metrizability of $M$-spaces, Proc. Japan Acad., 40, 176–179.
