Take the tour ×   Let $(X,x)$ be a pointed topological space.
Then the fundamental group $\pi_1(X,x)$ becomes a topological
space: Endow the set of maps $S^1 \to X$ with the compact-open
topology, endow the subset of maps mapping $1 \to x$ with the
subspace topology, and finally use the quotient topology on
$\pi_1(X,x)$. This topology is relevant in some situations. A
very interesting paper dealing with this topology is:  [1]
Daniel K. Biss, A Generalized Approach to the Fundamental Group,
The American Mathematical Monthly, Vol. 107  You can find this
online. This is somehow an introduction to  [2] Daniel K. Biss,
The topological fundamental group and generalized covering
spaces , Topology and its Applications, Vol. 124   How can we
prove that $\pi_1(X,x)$ is a topological group? Clearly the
inversion map $\pi_1(X,x) \to \pi_1(X,x)$ is continuous, since
$S^1 \to S^1, z \mapsto \overline{z}$ is continuous and induces
this map. But I don't know how to attack the continuity of the
multiplication. It's not hard t