Fundamental group

Let $A\subset \mathbb{C}$ be a path-connected open set, and fix a base point $z_0\in A$.
Consider the set of all closed curves in $A$ based at $z_0$, i.e., continuous maps $\gamma :[0,1]\to A$ with $\gamma (0)=\gamma (1)=z_0$. Call this set $\mathrm{\Lambda }(A;z_0)$.
Define a relation $\sim$ on $\mathrm{\Lambda }(A;z_0)$ by:

are homotopic as closed curves in $A$.

This is an equivalence relation. The fundamental group of $A$ at $z_0$ is the quotient set

equipped with the binary operation induced by concatenation of loops:

where ${\gamma }_0\ast {\gamma }_1$ traverses ${\gamma }_0$ then ${\gamma }_1$ at double speed. This operation is well-defined on homotopy classes, associative, has identity the constant loop at $z_0$, and inverses via reversal.

If is path-connected, ${\pi }_1(A;z_0)$ is independent of $z_0$ (up to isomorphism) and denoted simply ${\pi }_1(A)$. A set $A$ is simply connected if and only if ${\pi }_1(A)=\{e\}$ (the trivial group).