Path Homotopy is Equivalence Relation

Theorem
Let $X$ be a topological space.

Let $p, q \in X$.

Then path homotopy on the set of all paths in $X$ from $p$ to $q$ is an equivalence relation.

Proof
Suppose $X$ is a topological space.

Let $p, q \in X$.

Let $\sim$ denote the path homotopy on the set of all paths in $X$ from $p$ to $q$.

Checking in turn each of the criteria for equivalence:

Reflexivity
Thus $\sim$ is seen to be reflexive.

Symmetric
Thus $\sim$ is seen to be symmetric.

Transitive
Thus $\sim$ is seen to be transitive.

$\sim$ has been shown to be reflexive, symmetric and transitive.

Hence the result.