my favorite algebraic topology fact:
a topological group is a topological space with a group structure, ie an associative binary operation, an identity element, and inverses.

if X is a topological group with identity i, then the fundamental group pi_1(X,i) is always abelian

if there is interest i’ll do a thread on why this is the case. this is one of those things that i found genuinely surprising the first time i read it and still surprising when i went through the proof.

ok let& #39;s go!

the first thing that you might be thinking is that it& #39;s not terribly surprising that abelian groups show up somewhere with fundamental groups. after all, all the higher homotopy groups are abelian!

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so when i first learned it, the proof was given as an exercise, broken down into four parts:
for a topological group X with identity i, let \Omega(X,i) be the set of loops in X based at i. You can define a binary operation on the loop space by multiplying "pointwise".

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