Path traversal lemma

Let $𝛼 : 𝑥 ⇝ 𝑦$ be a continuous path and $𝜙 : [0, 1] \to [0, 1]$ be a continuous function with $𝜙 (0) = 0$ and $𝜙 (1) = 1$ . Then $𝛼 \circ 𝜙$ is a continuous path homotopic to $𝛼$ . #m/thm/homotopy

Proof

Let $𝛼 𝑠 (𝑡) = 𝛼 ((1 - 𝑠) 𝑡 + 𝑠 𝜙 (𝑡))$ . Then $𝛼 0 (𝑡) = 𝛼 (𝑡)$ and $𝛼 1 (𝑡) = 𝛼 \circ 𝜙 (𝑡)$ . Additionally, $𝛼 𝑠 (0) = 𝛼 (0)$ and $𝛼 𝑠 (1) = 𝛼 (1)$ . Hence $A : (𝑡, 𝑠) \mapsto 𝛼 𝑠 (𝑡)$ is a homotopy of paths.

#state/tidy | #lang/en | #SemBr