Homotopy class

Definition

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

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

• Reflexive: [ilmath]\alpha\simeq\alpha\text{ rel}\{0,1\} [/ilmath]
• Symmetric: [ilmath]\alpha\simeq\beta\text{ rel}\{0,1\}\implies \beta\simeq\alpha\text{ rel}\{0,1\} [/ilmath]
• Transitive: [ilmath]\alpha\simeq\beta\text{ rel}\{0,1\}\wedge\beta\simeq\gamma\text{ rel}\{0,1\}\implies \alpha\simeq\gamma\text{ rel}\{0,1\} [/ilmath]

See also

• Fundamental group