Prove by induction on dimension the following facts about the homology of a finite-dimensional CW complex X, using the observation that Xn/Xn−1 is a wedge sum of n-spheres:
(a) If X has dimension n then Hi(X)=0 for i>n and Hn(X) is free.
(b) Hn(X) is free with basis in bijective correspondence with the n-cells if there are no cells of dimension n−1 or n+1.
(c) If X has kn-cells, then Hn(X) is generated by at most k elements.