Reverse mathematics of regular countable second countable spaces

TL;DR

This paper explores the reverse mathematics of regular countable second countable spaces, establishing equivalences with various comprehension axioms and characterizing spaces like metrizable and well-orderable ones within subsystems of second-order arithmetic.

Contribution

It provides new reverse mathematical characterizations of topological properties of $CSCS$ spaces, linking them to comprehension axioms such as $ extbf{ACA}_0$ and $ extbf{ATR}_0$.

Findings

Arithmetic comprehension characterizes metrizable $T_3$ $CSCS$ spaces.

Every zero-dimensional separable $CSCS$ is homeomorphic to a linear order in $ extbf{ACA}_0$.

Certain topological properties are equivalent to $ extbf{Pi}^1_1$ comprehension.

Abstract

We study the reverse mathematics of characterization theorems of regular countable second countable spaces (or $CSCS$ for short). We prove that arithmetic comprehension is equivalent over $\textbf{RCA}_0$ to every $T_3$ $CSCS$ being metrizable, and we characterize the $T_3$ spaces which are metrizable over $\textbf{RCA}_0$. We show that Lynn's theorem for $CSCS$ can be carried out in $\textbf{ACA}_0$, namely that every zero dimensional separable space is homeomorphic to the order topology of a linear order. We also show that arithmetic comprehension is equivalent to every $T_2$ compact $CSCS$ being well-orderable. From general topology, we know that the locally compact $T_2$ $CSCS$ are the well-orderable $CSCS$, and that the $T_3$ scattered $CSCS$ are the completely metrizable $CSCS$. We show that these characterizations and a few others are equivalent to arithmetic transfinite recursion over $\textbf{RCA}_0$. We also find a few statments that are equivalent to $\Pi^1_1$ comprehension. In particular we show that every $T_3$ $CSCS$ has a Cantor Bendixson rank and that every $T_3$ $CSCS$ is the disjoint union of a scattered space and dense in itself space are equivalent to $\Pi^1_1$ comprehension over $\textbf{RCA}_0$.