I'm studying the fundamental group of a topological space and I've studied a proof checking that $\pi_1(X)$ is a group.
I'm thinking if the space of path components $\pi_0(X)$ is a group or not. I've tried to proof it is but don't see the way, so I suspect it isn't a group. Could anyone provide me a counterexample of this fact? (or tell me where could I found a proof in case it is).
The operation $*$ is the usual composition of pahts.
Thank you very much for your ideas.
In general, $\pi_0(X, x_0)$ is a pointed set. It does not have a natural group structure. However, if $X$ is a topological group, then $\pi_0(X, x_0)$ (where $x_0$ is the identity) inherits a natural group structure. Let $C(x)$ be the path component of $x \in X$. Then $C(x) * C(y) = C(x * y)$ is a well-defined operation on $\pi_0(X, x_0)$ that gives a group structure, with $C(x_0)$ being the identity. Try to prove this!
As for your question, since $\pi_0(X, x_0)$ is a set of path components, it cannot be given a group structure via path composition. How can you use path composition to "compose" two path components?