First Fundamental Group of 1-Sphere

Theorem

Let $\mathbb S^1$ be the $1$-sphere.

Let $\struct {\map {\pi _1} {\mathbb S^1}, \ast}$ be the first fundamental group of $\mathbb S^1$.

Let $\struct {\Z, +}$ be the additive group of integers.

Then $\struct {\map {\pi _1} {\mathbb S^1}, \ast}$ is isomorphic to $\struct {\Z, +}$.

Proof

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