homotopy

Overview

Two continuous maps are homotopic if (informally) we can continuously deform one into the other. We do this by adding another parameter that varies along an interval that “shows” the deformation.

Definition

For two continuous maps $f,g:X\to Y$, we say $f$ is homotopic to $g$ if there is a continuous map

$F:X\times I\to Y$

Such that $F(x,0)=f(x)\text{and}F(x,1)=g(x)$

We call $F$ a homotopy.