Path Homotopy is Equivalence Relation

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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 $q$.


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Checking in turn each of the criteria for equivalence:


Reflexivity

Path Homotopy is Reflexive


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Thus $\sim$ is seen to be reflexive.

$\Box$


Symmetric

Path Homotopy is Symmetric


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Thus $\sim$ is seen to be symmetric.

$\Box$


Transitive

Path Homotopy is Transitive


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Thus $\sim$ is seen to be transitive.

$\Box$


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

Hence the result.

$\blacksquare$


Sources

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