The Ginsburg--Sands theorem and computability theory

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 strict hierarchy generalizing the stable Ramsey's theorem for pairs (SRT²₂). We show that one of these, the Σ⁰₂ subset principle (Σ⁰₂-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.