A question on Pixley-Roy topology Let $R$ be Real line and let $F[R]$ be $\{x\subset R:\text{is finite}\}$ with Pixley-Roy topology.
Definition of Pixley-Roy topology is this: Basic neighborhoods of $F\in F[R]$ are the sets
$$[F,V]=\{H\in F[R]; F\subseteq H\subseteq V\}$$
for open sets $V\supseteq F$, see e.g. here.
Is it submetrizable? 
Thanks for your help:)
 A: Let $\mathscr{F}$ be the Pixley-Roy space in question; then $\mathscr{F}$ is submetrizable.
If $U\subseteq\Bbb R$ is open, let $U^+=\{F\in\mathscr{F}:F\cap U\ne\varnothing\}$ and $U^-=\{F\in\mathscr{F}:F\subseteq U\}$. If $F\in U^-$, then $F\in[F,U]\subseteq U^-$, and if $F\in U^+$, then $F\in[F\cap U,\Bbb R]\subseteq U^+$, so $U^-$ and $U^+$ are open in $\mathscr{F}$. Let $$\mathscr{S}=\{U^-:U\text{ is open in }\Bbb R\}\cup\{U^+:U\text{ is open in }\Bbb R\}\;;$$ clearly $\mathscr{S}$ is a subbase for a topology $\tau$ coarser than the Pixley-Roy topology on on $\mathscr{F}$. In fact $\tau$ is easily seen to be the Vietoris topology on $\mathscr{F}$. It’s well-known that if $X$ is metrizable, the Vietoris topology on the space $\mathscr{K}(X)$ of non-empty compact subsets of $X$ is metrizable; a proof can be found in this PDF. $\langle\mathscr{F},\tau\rangle$ is a subspace of $\mathscr{K}(\Bbb R)$ with the Vietoris topology, so $\langle\mathscr{F},\tau\rangle$ is metrizable, and $\mathscr{F}$ is submetrizable.
