The cross product:

This construction lies at the heart of the proof of Kunneth formula which relates the homology groups of $X\times Y$ in terms of the homologies of $X$ and $Y$. The first step would be to relate the singular chain complex of $X\times Y$ with those of $X$ and $Y$. This construction will be carried out naturally. Given a zero simplex $x \in X$ and a $q$ simplex $\sigma: \Delta_q\longrightarrow Y$ in $Y$, $x\times \sigma$ denotes the singular $q$ simplex in $X\times Y$ given by

$$x\times \sigma: \Delta_q\longrightarrow X\times Y$$

$$t\mapsto(x, \sigma(t)).\phantom{}$$

Likewise given a $q$ simplex $\tau$ in $X$ and a zero simplex $y$ in $Y$, one defines a $q$ simplex $\tau\times y$ in $X\times Y$. For a pair of singular simplices $\sigma\in \Delta_p(X)$ and $\tau \in \Delta_q(Y)$ we call $p+q$ the total degree of the pair $(\sigma, \tau)$.