A map \(f:X \to Y\) is homotopic to a map \(g:X \to Y\) if there is a homotopy \(F: X \times I \to Y\) with \(f = f_{0}\) and \(g = f_{1}\). \(F\) is “homotopy from \(f\) to \(g\)”. This induces a family of maps \(f_{t}(x) := F(x,t)\).

If \(F\) is a homotopy from \(f\) to \(g\) then the map \(\overline{F}:X \times I \to Y\) given by \[ (x,y) \maps \to F(x, 1-t) \] is a homotopy from \(g\) to \(f\). When \(f\) is homotopic to \(g\), write \(f \simeq g\). This is an equivalence relation.

A map \(f:X \to Y\) is called nullhomotopic if it is homotopic to a constant map.

A map \(f:X \to Y\) is called a homotopy equivalence if there is a map \(g:Y \to X\) where \(fg\) and \(gf\) are homotopic to identity maps. Write \(X \simeq Y\) if there exists a homotopy equivalence from \(X\) to \(Y\). This is an equivalence relation.

A retraction from \(X\) to \(A\nothing\) is a map \(r:X \to A\) with \(r\restrict_{A} = 1\). Note: \(r^2 = r\), so similar to a projection.

A deformation retraction from \(X\nothing\) to \(A\nothing\) is a homotopy from \(1:X \to X\) to a retraction from \(X\nothing\) to \(A\nothing\).