# Definition Let $X \subset \mathbb{R}^3$, such that $X$ is [[Homeomorphism|homeomorphic]] to a polygon $K$. Then the *Euler characteristic* $\chi(X)$ of $X$ is defined as: $ \begin{align} \chi(X) &= (\text{number of vertices in $K$}) ,\\ &- \text{(number of edges in $K$)},\\ &+ \text{(number of faces in $K$)} \end{align} $ In terms of the [[Connected Sum]] of two surfaces $X$ and $Y$: $ \chi(X \sharp Y) = \chi(X) + \chi(Y) - 2 $ # Examples $ \chi(\Sigma_g) = 2 - 2g $ where $g$ is the [[genus]] of the $g$-[[g-Handled Torus|handled torus]] $\Sigma_g$.