Proof That Continuous Functions From a Topological Space To The Real Numbers With The Euclidean Topology Are Homotopic
Theorem
Ifand
are continuous, then
and
are homotopic.
Proof
Letand
be continuous functions.
and
are said to be homotopic if
can be 'morphed' onto
and vice versa, or more concisely, if a continuous function
exists such that
and
Ifdefine
then
and
are homotopic.