A universal coregular countable second-countable space

A Hausdorff topological space $X$ is called $\mathit{\text{superconnected}}$ (resp. $\mathit{\text{coregular}}$) if for any nonempty open sets $U_1,\dots U_n\subseteq X$, the intersection of their closures ${\bar{U}}_1\cap \cdots \cap {\bar{U}}_n$ is not empty (resp. the complement $X\setminus ({\bar{U}}_1\cap \cdots \cap {\bar{U}}_n)$ is a regular topological space). A canonical example of a coregular superconnected space is the projective space $\mathbb{Q}{\mathsf{P}}^{\mathrm{\infty }}$ of the topological vector space $\mathbb Q^{<\omega}={(x_n)_{n\in\omega}\in \mathbb Q^{\omega}:|{n\in\omega:x_n\ne 0}|<\omega}$overthefieldofrationals$\mathbb Q$\mathrm{.}Thespace$\mathbb Q\mathsf P^\infty$isthequotientspaceof$\mathbb Q^{<\omega}\setminus{0}^\omega$bytheequivalencerelation$x\sim y$iff$\mathbb Q{\cdot}x=\mathbb Q{\cdot}y$. We prove that every countable second-countable coregular space is homeomorphic to a subspace of $\mathbb{Q}{\mathsf{P}}^{\mathrm{\infty }}$, and a topological space $X$ is homeomorphic to $\mathbb{Q}{\mathsf{P}}^{\mathrm{\infty }}$ if and only if $X$ is countable, second-countable, and admits a decreasing sequence of closed sets $(X_n{)}_{n\in \omega }$ such that (i) $X_0=X$, ${\bigcap }_{n\in \omega }{X}_n=\mathrm{\varnothing }$, (ii) for every $n\in \omega$ and a nonempty open set $U\subseteq X_n$ the closure $\bar{U}$ contains some set $X_m$, and (iii) for every $n\in \omega$ the complement $X\setminus X_n$ is a regular topological space. Using this topological characterization of $\mathbb Q\mathsf P^\infty$wefindtopologicalcopiesofthespace$\mathbb Q\mathsf P^\infty$ among quotient spaces, orbit spaces of group actions, and projective spaces of topological vector spaces over countable topological fields.