Speaker: Damir Dzhafarov
University of Connecticut
Will give a Logic Seminar entitled:
The Ginsburg-Sands theorem and computability`
Abstract: In their 1979 paper ``Minimal Infinite Topological Spaces'', Ginsburg and Sands proved that every infinite topological space has an infinite subspace homeomorphic to exactly one of the following five topologies on $\omega$: indiscrete, discrete, initial segment, final segment, and cofinite. The proof, while nonconstructive, features an interesting application of Ramsey's theorem for pairs ($\mathsf{RT}^2_2$). 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 $\mathsf{ACA}_0$, while for Hausdorff spaces it is provable in $\mathsf{RCA}_0$. 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 ($\mathsf{CAC}$). The most surprising case is that of the Ginsburg-Sands theorem restricted to $T_1$ spaces. Here, we show that the principle lies strictly between $\mathsf{ACA}_0$ and $\mathsf{RT}^2_2$, yielding perhaps the first natural theorem of ordinary mathematics (i.e., conceived outside of logic) to occupy this interval. I will discuss the proofs of both the implications and separations, which feature several novel combinatorial elements, and survey a new class of purely combinatorial principles below $\mathsf{ACA}_0$ and not implied by $\mathsf{RT}^2_2$ revealed by our investigation. This is joint work with Heidi Benham, Andrew DeLapo, Reed Solomon, and Java Darleen Villano.
Date: 04-18-2024
Time: 2:00 pm
Location: 258 Hayes-Healy Bldg