1. The problem statement, all variables and given/known data Prove that if Y is contractible, then any two maps from X to Y are homotopic. I feel like I have a very, very, very sloppy proof :( 2. Relevant equations 3. The attempt at a solution Assume Y is contractible to a point y0 held fixed, then there is a map F:Y x [0,1] -> Y such that (i) F(y,0)=y0, for all y in Y (ii) F(y,1)=y, for all y in Y (iii) F(y0,t)=y0, for all t in [0,1] Let F(y,t) be the homotopy of Y to a point y0 Claim: any map f:X -> Y is homotopic to y0 by the homotopy ft(x)=F(f(x),t) Proof of Claim: Since f(x)=y in Y, ft=F(y,t), where y is in Y and t is in [0,1], And since Y is contractible, f:X -> Y is homotopic to y0 Therefore, any two maps from X to Y are homotopic.