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Proof That The Property of Being Homotopic is an Equivalence Relation

Theorem

The property of being homotopic is an equivalence relation.

Proof

1. f sim f . Let h:X times [0,1] rightarrw Y be defined by h(x,t)=f(x) then h(x,0)=h(x,1)=f(x) so f sim f.

2. f sim g. Sincethere is a homotopywith

Defineby

Thenand

Sinceis continuous, so isand

3.then

Sincethere issuch that

Sincethere issuch that

Define

Thenandso

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