Tietze extension does not always work in constructive mathematics if closed sets are defined as sequentially closed sets

TL;DR

This paper demonstrates that the classical Tietze Extension theorem fails in constructive mathematics when closed sets are defined via sequential closure, using a specific counterexample involving an unextendible function.

Contribution

It provides a counterexample showing Tietze Extension does not always hold in constructive mathematics under sequentially closed set definitions.

Findings

Tietze Extension fails in constructive setting with sequentially closed sets

Constructed unextendible function on a discrete metric space

Contradiction arises if Tietze extension is assumed to hold

Abstract

We prove that Tietze Extension does not always exist in constructive mathematics if closed sets on which the function we are extending are defined as sequentially closed sets. Firstly, we take a discrete metric space as our topological space. Now all sets open and sequentially closed. Then, we form an unextendible algorithmic function transforming positive integers to 0 and 1, looking at the preimages of these values as our sequentially closed sets. Then we show that if the Tietze theorem conclusion holds for these closed sets then the unextendible function is extendible thus giving us a contradiction.