I'm totally stuck on these two. The first is... Let A be a subset of X; suppose r:X->A is a continuous map from X to A such that r(a)=a for each a e A. If a_0 e A, show that... r* : Pi_1(X,a_0) -> Pi_1(A,a_0) ...is surjective. Note: Pi_1 is the first homotopy group and r* is the homomorphism induced by h. I can visually see in my mind why this is so, but I can't even think of how to write this down at all. I'm still thinking about it. No need to post anything right now. The way I'm thinking that is if [f] is in A then I need to show that there is a g in X such that g is path homotopic to f using probably r to create my path homotopy. (f is a loop around a_0 in A) Once I do that, then it should come out like... r([g]) = [r o g] = [f]. I'm still thinking about this.