A recent paper by Benham, DeLapo, Dzhafarov, Solomon, and Villano entitled "Ginsburg-Sands theorem and computability theory" analyzes computability theoretical and reverse mathematical strength of a topological theorem by Ginsburg and Sands, along with several weakened versions. The original theorem states that every infinite topological space has an infinite subspace homeomorphic to one of the following on the natural numbers: indiscrete, initial segment, final segment, discrete, or cofinite. In this original paper, it is claimed that the theorem is a consequence of Ramsey's Theorem, and though it has been shown by Benham, DeLapo, Dzhafarov, Solomon, and Villano that the full theorem is equivalent over RCA_0 to ACA_0, there is a weakened version that is equivalent over RCA_0 to CAC (Chain-antichain Principle), a consequence of Ramsey's Theorem. One interesting feature of the proof of this equivalence is that, not only an application CAC, but also an application of ADS (Ascending/descending Sequence Principle), which is a consequence of CAC, is used. This inspires the question of whether this weakened version of the Ginsburg-Sands Theorem and CAC, when viewed as problems, are Weihrauch equivalent.
This talk is a continuation of part of the lecture series given by Damir Dzhafarov and will provide a more detailed overview of some of the proofs related to the Weihrauch degree of the weakened Ginsburg–Sands Theorem with the closure operator.