Homotopy class

Definition

The relation of paths being end-point-preserving homotopic is an Equivalence relation

That is $\alpha\simeq\beta\text{ rel}\{0,1\}$ where $\alpha$ and $\beta$ are paths from $a$ to $b$ (which are not necessarily distinct as it may be a loop) is an equivalence relation, which is to say:

• Reflexive: $\alpha\simeq\alpha\text{ rel}\{0,1\}$
• Symmetric: $\alpha\simeq\beta\text{ rel}\{0,1\}\implies \beta\simeq\alpha\text{ rel}\{0,1\}$
• Transitive: $\alpha\simeq\beta\text{ rel}\{0,1\}\wedge\beta\simeq\gamma\text{ rel}\{0,1\}\implies \alpha\simeq\gamma\text{ rel}\{0,1\}$

See also

• Fundamental group