Definable Tietze extension property in o-minimal expansion of ordered group

TL;DR

This paper establishes an equivalence in o-minimal expansions of ordered groups between the existence of a definable bijection linking bounded and unbounded intervals and the definable extension property for continuous functions, highlighting a key structural feature.

Contribution

It proves the equivalence between the definable Tietze extension property and the existence of a definable bijection between bounded and unbounded intervals in o-minimal expansions of ordered groups.

Findings

Definable bijection exists between bounded and unbounded intervals.

Definable continuous functions extend over closed sets.

Equivalence of extension property and interval bijection in o-minimal structures.

Abstract

The following two assertions are equivalent for an o-minimal expansion of an ordered group $\mathcal M=(M,<,+,0,\ldots)$. There exists a definable bijection between a bounded interval and an unbounded interval. Any definable continuous function $f:A \rightarrow M$ defined on a definable closed subset of $M^n$ has a definable continuous extension $F:M^n \rightarrow M$.