Notation in Fundamental Groups in Topological Spaces

Question

I was reading about Fundamental Groups in Topological Spaces at WikiPedia. In the middle of the text, they define the following set $\pi_1(X,x_0):$

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I guess I don't understand the meaning of "$\text{/homotopy}$". What does this mean?

Answer 1

If you have a set $S$ and an equivalence relation $\sim$ on $S$, then $$S / \sim$$ denotes the set of all equivalence classes with respect to $\sim $. In your question we have $S =$ set of all loops $\gamma : [0,1] \to X$ based at $x_0$. You have the relation $\gamma$ is homotopic to $\gamma'$ which is an equivalence relation on $S$. If you write $\gamma \simeq \gamma'$ for this equivalence relation, then $$\pi_1(X,x_0) = S/\simeq \phantom{.}= S/\text{homotopy} .$$