The reverse mathematics of the Tietze extension theorem

TL;DR

This paper establishes the logical strength of various forms of the Tietze extension theorem within reverse mathematics, showing their equivalence to WKL_0 over RCA_0, thus clarifying their foundational significance.

Contribution

It proves the equivalence of multiple versions of the Tietze extension theorem with WKL_0 in reverse mathematics, confirming a conjecture and answering an open question.

Findings

Several versions of the Tietze extension theorem are equivalent to WKL_0 over RCA_0.

The results confirm a conjecture by Giusto and Simpson.

The work addresses an open question in reverse mathematics.

Abstract

We prove that several versions of the Tietze extension theorem for functions with moduli of uniform continuity are equivalent to WKL_0 over RCA_0. This confirms a conjecture of Giusto and Simpson that was also phrased as a question in Montalb\'an's "Open questions in reverse mathematics."