Hybrid Maximal Filter Spaces

TL;DR

This paper introduces hybrid maximal filter spaces, a new encoding of topology in second order arithmetic, enabling coding of all second countable MF spaces and simplifying metrization proofs within weaker logical systems.

Contribution

It proposes hybrid MF spaces as a modification of proper MF spaces, allowing broader topological coding and proving a metrization theorem in a weaker logical framework.

Findings

Hybrid MF spaces can code any second countable MF space.

Metrization theorem for hybrid MF spaces proven in ACA_0.

Proper MF spaces are limited in coding topological spaces.

Abstract

We introduce a new way of encoding general topology in second order arithmetic that we call hybrid maximal filter (hybrid MF) spaces. This notion is a modification of the notion of a proper MF space introduced by Montalb\'an. We justify the shift by showing that proper MF spaces are not able to code most topological spaces, while hybrid MF spaces can code any second countable MF space. We then answer Montalb\'an's question about metrization of well-behaved MF spaces to this shifted context. To be specific, we show that in stark contrast to the original MF formalization used by Mummert and Simpson, the metrization theorem can be proven for hybrid MF spaces within $\text{ACA}_0$ instead of needing $\Pi_2^1-\text{CA}_0$.