Let (X, x 0) and (Y, y0) be given and let (X × Y, (x0, y0)) be their product.
(a) Show that for any (K, k0) that [K, X × Y ] = [K, X ] × [K, Y ]. In particular pi1(X × Y ) = π1(X ) × pi1(Y )
(b) This implies that if a is a loop in X and b a loop in Y and we let A(t) = (a(t), yo) and B(t) = (xo, b(t)) then A and B commute in pi1(X ×Y ). Show this by finding an explicit homotopy between AB and BA