Proposition

Let $X$ be a topological space. For any points $p, q \in X$, path homotopy is an equivalence relation on the set of all paths in $X$ from $p$ to $q$.

Solution

Let $f, g, h$ be paths in $X$ from $p$ to $q$.

Reflexive property

$F(s, t) = f(s)$ is a path homotopy from $f$ to $f$. Thus $f$ is path-homotopic to $f$.

Symmetric property

Suppose $f$ is path-homotopic to $g$. Let $F: I \times I \rightarrow X$ be a path homotopy from $f$ to $g$. Then consider $G(s, t) = F(s, 1 - t)$.

• $G$ is continuous because
  • $t \mapsto 1 - t$ is continuous.
  • $s \times t \mapsto s \times (1 - t)$ is continuous since it is a product of two continuous functions.
  • A composition of continuous functions is continuous.
• $G(s, 0) = F(s, 1) = g(s)$.
• $G(s, 1) = F(s, 0) = f(s)$.
• $G(0, t) = F(0, 1 - t) = p$.
• $G(1, t) = F(1, 1 - t) = q$.

Therefore, $G$ is a path homotopy from $g$ to $f$, so $g$ is path-homotopic to $f$.

Transitive property

Suppose $f$ is path-homotopic to $g$ and $g$ is path-homotopic to $h$. Let $F$ be a path homotopy from $f$ to $g$, and $G$ be a path homotopy from $g$ to $h$. Then consider $H: I \times I \rightarrow X$ such that

• Is $H$ well-defined?
  • $\forall s \in I, H(s, 1/2) = F(s, 1) = g(1) = G(s, 0) = H(s, 1/2)$.
• $H$ is continuous by the gluing lemma.
• $H(s, 0) = F(s, 0) = f(s)$.
• $H(s, 1) = G(s, 1) = h(s)$.
• $H(0, t) = F(0, 2t) = p$.
• $H(1, t) = G(1, 2t - 1) = q$.

Therefore, $H$ is a path homotopy from $f$ to $h$, so $f$ is path-homotopic to $h$.

Hence, path homotopy is an equivalence relation.