The Ginsburg–Sands theorem and computability theory

Abstract

The Ginsburg–Sands theorem from topology states that every infinite topological space has an infinite subspace homeomorphic to exactly one of the following five topologies on ω: indiscrete, discrete, initial segment, final segment, and cofinite. The original proof is nonconstructive, and features an interesting application of Ramsey's theorem for pairs (RT22). We analyze this principle in computability theory and reverse mathematics, using Dorais's formalization of CSC spaces. Among our results are that the Ginsburg–Sands theorem for CSC spaces is equivalent to ACA0, while for Hausdorff spaces it is provable in RCA0. Furthermore, if we enrich a CSC space by adding the closure operator on points, then the Ginsburg–Sands theorem turns out to be equivalent to the chain/antichain principle (CAC). The most surprising case is that of the Ginsburg–Sands theorem restricted to T1 spaces. Here, we show that the principle lies strictly between ACA0 and RT22, yielding arguably the first natural theorem from outside logic to occupy this interval. As part of our analysis of the T1 case we introduce a new class of purely combinatorial principles below ACA0 and not implied by RT22 which form a hierarchy generalizing the stable Ramsey's theorem for pairs (SRT22). We show that one of these, the Σ20 subset principle (Σ20-Subset), has the property that it, together with the cohesive principle (COH), is equivalent over RCA0 to the Ginsburg–Sands theorem for T1 CSC spaces.