Euler characteristic of product is product of Euler characteristics

Statement

For two spaces

Suppose $X$ and $Y$ are (possibly homeomorphic/equal) topological spaces and each of them is a Space with finitely generated homology (?). Then, the product of topological spaces $X\times Y$ is also a space with finitely generated homology and the Euler characteristic (?)s $\chi (X)$, $\chi (Y)$, 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 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 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.

Related facts

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.