The fundamental group of the circle

Step 1 — Set up the covering map. Let $p: \mathbb{R} \to S^1$ be $p(t) = (\cos 2\pi t, \sin 2\pi t)$, a continuous surjection where every point of $S^1$ has an evenly-covered neighborhood (a small arc's preimage under $p$ is a disjoint union of open intervals in $\mathbb{R}$, each mapped homeomorphically onto the arc). Fix basepoint $1 = p(0) \in S^1$.

Step 2 — Path lifting. Given any loop $\gamma: [0,1] \to S^1$ with $\gamma(0)=\gamma(1)=1$, there is a unique continuous lift $\tilde\gamma: [0,1] \to \mathbb{R}$ with $p \circ \tilde\gamma = \gamma$ and $\tilde\gamma(0) = 0$ (construct it by covering $[0,1]$ with finitely many subintervals on which $\gamma$ stays inside one evenly-covered neighborhood, and lift piece by piece, each time choosing the branch of $p^{-1}$ continuing from the previous endpoint). Define $\deg(\gamma) = \tilde\gamma(1) \in \mathbb{Z}$ (an integer since $p(\tilde\gamma(1)) = \gamma(1) = 1 = p(0)$ forces $\tilde\gamma(1) \in \mathbb{Z}$).

Step 3 — Homotopy invariance. If $\gamma \simeq_p \gamma'$ via homotopy $H$, the homotopy lifting property (proved the same way as path lifting, one strip at a time) gives a continuous lift $\tilde H$ with $\tilde H(s,0)=\tilde\gamma(s)$, and $\tilde H(0,t)=0$ for all $t$ (constant, since it lifts the constant basepoint loop). Then $t \mapsto \tilde H(1,t)$ is a continuous integer-valued function of $t$ (by the same argument as Step 2), hence constant; so $\deg(\gamma) = \tilde H(1,0) = \tilde H(1,1) = \deg(\gamma')$. Thus $\deg$ is well-defined on homotopy classes $[\gamma] \mapsto \deg(\gamma)$.

Step 4 — Homomorphism property. For loops $\gamma, \delta$, lifting $\gamma * \delta$ by first lifting $\gamma$ to end at $\deg(\gamma)$, then lifting $\delta$ starting from $\deg(\gamma)$ (a shifted copy of $\delta$'s lift, valid since $p$ is invariant under integer translation) shows $\deg(\gamma * \delta) = \deg(\gamma) + \deg(\delta)$. So $[\gamma] \mapsto \deg(\gamma)$ is a group homomorphism $\pi_1(S^1,1) \to \mathbb{Z}$.

Step 5 — Bijectivity. Surjective: for any $n \in \mathbb{Z}$, $\gamma_n(s) = p(ns)$ is a loop with $\deg(\gamma_n)=n$. Injective: if $\deg(\gamma)=0$, the lift $\tilde\gamma$ is a loop in $\mathbb{R}$ based at $0$ (since $\tilde\gamma(1)=0=\tilde\gamma(0)$); $\mathbb{R}$ is convex so the straight-line homotopy $\tilde H(s,t) = (1-t)\tilde\gamma(s)$ contracts $\tilde\gamma$ to the constant path at $0$ rel endpoints, and composing with $p$ gives a path-homotopy from $\gamma$ to the constant loop, so $[\gamma]$ is trivial. Hence $\deg$ is a bijective homomorphism, i.e. an isomorphism $\pi_1(S^1,1) \cong \mathbb{Z}$.