Nonexistence of Algorithm Deciding if the Fundamental Group of an Arbitrary Constructive Compact Set is Trivial

In this paper, we give an introduction to the compact space under Constructive Mathematics, ultimately showing that there exist no computer programs that can always determine if a fundamental group of a constructive compact topological space is trivial or not. We will begin with the non-halting case of the programs, which shows that the fundamental group of our selected set is trivial. From here, we will explore the halting case of the programs, which can be solved by arbitrarily picking a nontrivial subset of our selected set. The main result of this section is that we can show that the algorithm is decidable due to our initial construction, while realizing its contradiction with our input. We want to show that such an algorithm doesnt exist by employing the proof-by-contradiction method.