# Confusion regarding CW complex structure of product

I find that if $$X,Y$$ are CW complexes then $$X \times Y$$ is a CW complex with skeleta $$(X \times Y)^{(n)}=\cup_{p+q=n} X^{(p)} \times Y^{(q)}$$ .

So while considering $$S^n$$ with CW complex structure:

$$X^0=*$$

$$X^1=*$$

…..

$$X^{n-1} = *$$

$$X^{n} = S^n$$

$$X^{n+1} = S^n$$

….

What are the skeleta of $$S^5 \times S^7$$ ?

By the above formula I have found it seems like,

$$(S^5\times S^7)^{0}=*$$

$$(S^5\times S^7)^{1}=(S^5)^{(0)} \times (S^7)^{(1)} \cup (S^5)^{(1)} \times (S^7)^{(0)}=* \cup *$$

$$(S^5\times S^7)^{2}=(S^5)^{(0)} \times (S^7)^{(2)} \cup (S^5)^{(1)} \times (S^7)^{(1)} \cup (S^5)^{(2)} \times (S^7)^{(0)}=* \cup * \cup *$$

….

$$(S^5\times S^7)^{5}= S^5 \cup * \cup * \cup * \cup * \cup *$$

and so on ?

My aim is not to find in particular the CW structure of this product but to clear the confusion regarding what is going on. Can someone please illustrate this example and help me understand!