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Definition 4 (Definition: Fundamental Group) Let \(X\) be a topological space and \(x_0 \in X\) a basepoint. A loop based at \(x_0\) is a continuous map \(\gamma: [0,1] \to X\) with \(\gamma(0) = \gamma(1) = x_0\). Two loops \(\gamma, \delta\) are homotopic relative to \(x_0\) (written \(\gamma \simeq \delta\)) if there exists a continuous map \(H: [0,1] \times [0,1] \to X\) such that \[H(s,0) = \gamma(s), \quad H(s,1) = \delta(s), \quad H(0,t) = H(1,t) = x_0\] for all \(s,t \in [0,1]\). This is an equivalence relation; denote the equivalence class of \(\gamma\) by \([\gamma]\).
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