Theorem 33.1:

There exists a bilinear map

$$S_p(X)\times S_q(Y) \longrightarrow S_{p+q}(X\times Y)$$

$$(\sigma, \tau)\mapsto \sigma\times \tau,\phantom{XXXX}$$

with the following properties

(i)

For zero simplices $x \in X$, $y \in Y$ and singular simplices $\sigma:\Delta_p\longrightarrow X$ and $\tau:\Delta_q\longrightarrow Y$ the products $x\times \tau$, $\sigma\times y$ are already defined above.

(ii)

Naturality: Suppose that $f:X\longrightarrow X^{\prime}$ and $g:Y\longrightarrow Y^{\prime}$ are two continuous maps and $f\times g:X\times Y\longrightarrow X^{\prime}\times Y^{\prime}$ denotes the product map $(f\times g)(x, y) = (f(x), g(y))$, then

$$(f\times g)_{\sharp}(\sigma\times \tau) = f_{\sharp}(\sigma)\times g_{\sharp}(\tau) \eqno(33.1)$$

(iii)

Generalized Leibnitz' rule: If $\sigma\in S_p(X)$ and $\tau\in S_q(Y)$ then

$$\partial (\sigma\times \tau) = \partial \sigma\times \tau + (-1)^p (\sigma\times \partial \tau) \eqno(33.2)$$