Euler characteristic of connected Lie group is zero or one

Statement

Suppose $G$ is a connected Lie group. Then $G$ is either a Space with Euler characteristic zero (?) or a Space with Euler characteristic one (?). In other words, the Euler characteristic (?) of $G$ is either $0$ or $1$.

The case of Euler characteristic one occurs if and only if $G$ is contractible. Otherwise, $G$ has Euler characteristic zero.

More generally, the Euler characteristic of a Lie group is equal to either zero or the number of path components.

Facts used

1. Euler characteristic of compact connected nontrivial Lie group is zero