A universal coregular countable second-countable space

Abstract

A Hausdorff topological space $X$ is called $\textit{superconnected}$ (resp. $\textit{coregular}$) if for any nonempty open sets $U_1,\dots U_n\subseteq X$, the intersection of their closures $\bar U_1\cap\dots\cap\bar U_n$ is not empty (resp. the complement $X\setminus (\bar U_1\cap\dots\cap\bar U_n)$ is a regular topological space). A canonical example of a coregular superconnected space is the projective space $\mathbb Q\mathsf P^\infty$ of the topological vector space $\mathbb Q^{<\omega}=\{(x_n)_{n\in\omega}\in \mathbb Q^{\omega}:|\{n\in\omega:x_n\ne 0\}|<\omega\}$ over the field of rationals $\mathbb Q$. The space $\mathbb Q\mathsf P^\infty$ is the quotient space of $\mathbb Q^{<\omega}\setminus\{0\}^\omega$ by the equivalence relation $x\sim y$ iff $\mathbb Q{\cdot}x=\mathbb Q{\cdot}y$. We prove that every countable second-countable coregular space is homeomorphic to a subspace of $\mathbb Q\mathsf P^\infty$, and a topological space $X$ is homeomorphic to $\mathbb Q\mathsf P^\infty$ 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) $X_0=X$, $\bigcap_{n\in\omega}X_n=\emptyset$, (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 $\mathbb Q\mathsf P^\infty$ we find topological copies of the space $\mathbb Q\mathsf P^\infty$ among quotient spaces, orbit spaces of group actions, and projective spaces of topological vector spaces over countable topological fields.