This question is from Munkres Algebraic Topology section 25 question 6
The question, vertabim, says.
"We shall study the homology of $X\times Y$ in chapter 7.For the present, prove the following, assuming all the spaces involved are polyhedra.
(a) Show that if $p\in S^n$, $$H_{q}(X\times S^n,X\times p)\simeq H_{q-n}(X)\hspace{15pt}(1)$$
[Hint: Write $S^{n}$ as the union of its upper and lower hemispheres, and proceed by induction on n.]
(b) Show that if $p\in Y$, the homology exact sequence of $(X\times Y,X\times p)$ breaks up into short exact sequences that split
(c) Prove that $H_{q}(X\times S^n)\simeq H_{q-n}(X)\oplus H_{q}(X)$
(d) Compute the homology of $S^n\times S^m$
Note that this question is in the section of the textbook labelled "Mayer Vietoris" so I am expecting to use it somewhere.
My attempt: (a) Using the hint, I will write $S^n=E_{+}^n\cup E_{-}^n$ where $E_{+}^n$ and $E_{-}^{n}$ represent the upper and lower hemispheres of $S^n$ respectively. I will now use the relative Mayer Vietoris sequence using $K_{0}=X\times E_{+}^n$ and $K_{1}=X\times E_{-}^{n}$ Doing so will give me $K_{0}\cup K_{1}=X\times S^{n}$ which I shall call $K$ from here on in and that $K_{0} \cap K_{1}=X\times S^{n-1}$. I intend to use the relative Mayer Vietoris sequence (which is question 2 of the same section in Munkres Algebraic Topology). I may now write
$...\rightarrow H_{i}(X\times S^{n-1},L_{0}\cap L_{1})\rightarrow H_{i}(X\times E_{+}^n,L_{0})\oplus H_{i}(X\times E_{-}^{n},L_{1})\rightarrow H_{i}(X\times S^n,L_{0}\cup L_{1})\rightarrow...$
At this stage I think I need to choose $L_{0}$ and $L_{1}$ such that $L_{0}\cup L_{1}=X\times p$, but I'm not sure how to do that and have the desired result fall out. Another note, the hint also says I need to prove this by induction, For $n=0$ $S^{n}$ is just a point and hence $X\times S^n$ is the same as $X\times p$ and hence the left hand side of 1 should just be 0. I can't see why then $H_{q}(X)$ should be zero (What is stopping $X$ from being a $q$ dimensional simplex which would mean $H_{q}(X)=\mathbb{Z}$), thus I am also struggling to prove the base case of this question
This is all I have got so far, you need not answer all the questions, It would be great if you could help me with part a and possibly give small hints for parts b an c and leave me to do part d.