Let $X = K \times \mathbb{R}P^3$ be the product of the Klein bottle with the 3 -dimensional projective space.
1) Find a CW-decomposition of $X$.
2) Determine the chain complex $(C_{*}(X),\delta )$ associated to that cell decomposition.
Attempt: I know that a CW-decomposition of $K$ is $e^0\cup e^1\cup e^1\cup e^2$ and $\mathbb{R}P^3$ is $e^0\cup e^1\cup e^2\cup e^3$. Then we get a CW-decomposition of $X$ as
$(e^0\times e^0)\cup (e^0\times e^1)\cup (e^0\times e^2)\cup (e^0\times e^3)\cup (e^1\times e^0)\cup (e^1\times e^1)\cup (e^1\times e^2)\cup (e^1\times e^3)\cup (e^1\times e^0)\cup (e^1\times e^1)\cup (e^1\times e^2)\cup (e^1\times e^3)\cup (e^2\times e^0)\cup (e^2\times e^1)\cup (e^2\times e^2)\cup (e^2\times e^3)=e^0\cup e^1\cup e^2\cup e^3\cup e^1\cup e^2\cup e^3\cup e^4\cup e^1\cup e^2\cup e^3\cup e^4\cup e^2\cup e^3\cup e^4\cup e^5$
I am very new in homology theory. But I do not know how to determine the chain complex $(C_{*}(X),\delta )$. Thanks!