Euler characteristic of product is product of Euler characteristics

For two spaces
Suppose $$X$$ and $$Y$$ are (possibly homeomorphic/equal) topological spaces and each of them is a fact about::space with finitely generated homology. Then, the product of topological spaces $$X \times Y$$ is also a space with finitely generated homology and the fact about::Euler characteristics $$\chi(X)$$, $$\chi(Y)$$, and $$\chi(X \times Y)$$ are related as follows:

$$\chi(X \times Y) = \chi(X)\chi(Y)$$

In words, the Euler characteristic of the product is the product of the Euler characteristics.

For multiple spaces
Suppose $$X_1,X_2,\dots,X_n$$ are (possibly homeomorphic/equal) topological spaces and each of them is a fact about::space with finitely generated homology. Then, the product of topological spaces $$X_1 \times X_2 \times \dots \times X_n$$ is also a space with finitely generated homology and its fact about::Euler characteristic is given by:

$$\chi(X_1 \times X_2 \times \dots \times X_n) = \chi(X_1)\chi(X_2) \dots \chi(X_n)$$.

In words, the Euler characteristic of the product is the product of the Euler characteristics.

Corollaries

 * If $$X$$ is a space with finitely generated homology, $$\chi(X^n) = (\chi(X))^n$$ where $$X^n$$ is the $$n$$-fold product of $$X$$ with itself.
 * If $$X$$ is a space with zero Euler characteristic, and $$Y$$ is a space with finitely generated homology, then $$X \times Y$$ is also a space with zero Euler characteristic.
 * The Euler characteristic of $$X \times X$$ is always nonnegative.