# Definition:Fundamental Group

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## Definition

Let $\struct {X, x_0}$ be a pointed topological space with base point $x_0$.

The fundamental group $\map {\pi_1} {X, x_0}$ of $X$ at the base point $x_0$ is the set of homotopy classes of loops with base point $x_0$ with multiplication of homotopy classes of paths.

## Also known as

The fundamental group is also known more explicitly as the fundamental homotopy group.