The Non-Existence of Algorithm Deciding Whether a Constructive Compact Topological Space is Simply Connected

Abstract

This paper discusses the non-existence of an algorithm Q always deciding whether a constructive compact topological space is simply connected or not, that is whether its fundamental group is trivial or not. We construct a program R that generates compact topological spaces, then we use R and Q to get an extension of an unextendible function to get the contradiction and finish the proof.