A universal coregular countable second-countable space

Abstract

A Hausdorff topological space X is called superconnected (resp. coregular) if for any nonempty open sets U1,…Un⊆X, the intersection of their closures U‾1∩…∩U‾n is not empty (resp. the complement X∖(U‾1∩…∩U‾n) is a regular topological space). A canonical example of a coregular superconnected space is the projective space QP∞ of the topological vector space Q<ω={(xn)n∈ω∈Qω:|{n∈ω:xn≠0}|<ω} over the field of rationals Q. The space QP∞ is the quotient space of Q<ω∖{0}ω by the equivalence relation x∼y iff Q⋅x=Q⋅y.We prove that every countable second-countable coregular space is homeomorphic to a subspace of QP∞, and a topological space X is homeomorphic to QP∞ if and only if X is countable, second-countable, and admits a decreasing sequence of closed sets (Xn)n∈ω such that (i) X0=X, ⋂n∈ωXn=∅, (ii) for every n∈ω and a nonempty relatively open set U⊆Xn the closure U‾ contains some set Xm, and (iii) for every n∈ω the complement X∖Xn is a regular topological space. Using this topological characterization of QP∞ we find topological copies of the space QP∞ among quotient spaces, orbit spaces of group actions, and projective spaces of topological vector spaces over countable topological fields.