A universal coregular countable second-countable space

A Hausdorff topological space $X$ is called $textit{superconnected}$ (resp. $textit{coregular}$) if for any nonempty open sets $U_1,dots U_nsubseteq X$, the intersection of their closures $bar U_1capdotscapbar U_n$ is not empty (resp. the complement $Xsetminus (bar U_1capdotscapbar U_n)$ is a regular topological space). A canonical example of a coregular superconnected space is the projective space $mathbb Qmathsf P^infty$ of the topological vector space $mathbb Q^{<omega}={(x_n)_{ninomega}in mathbb Q^{omega}:|{ninomega:x_n e 0}|<omega}$ over the field of rationals $mathbb Q$. The space $mathbb Qmathsf P^infty$ is the quotient space of $mathbb Q^{<omega}setminus{0}^omega$ by the equivalence relation $xsim y$ iff $mathbb Q{cdot}x=mathbb Q{cdot}y$. We prove that every countable second-countable coregular space is homeomorphic to a subspace of $mathbb Qmathsf P^infty$, and a topological space $X$ is homeomorphic to $mathbb Qmathsf P^infty$ if and only if $X$ is countable, second-countable, and admits a decreasing sequence of closed sets $(X_n)_{ninomega}$ such that (i) $X_0=X$, $bigcap_{ninomega}X_n=emptyset$, (ii) for every $ninomega$ and a nonempty open set $Usubseteq X_n$ the closure $bar U$ contains some set $X_m$, and (iii) for every $ninomega$ the complement $Xsetminus X_n$ is a regular topological space. Using this topological characterization of $mathbb Qmathsf P^infty$ we find topological copies of the space $mathbb Qmathsf P^infty$ among quotient spaces, orbit spaces of group actions, and projective spaces of topological vector spaces over countable topological fields.