Proof That Continuous Functions From a Topological Space To The Real Numbers With The Euclidean Topology Are Homotopic

Theorem

Ifandare continuous, thenandare homotopic.

Proof

Letandbe continuous functions.andare said to be homotopic if can be 'morphed' ontoand vice versa, or more concisely, if a continuous functionexists such thatand

Ifdefinethenandare homotopic.