TITLE: Euler characteristic 1: Half a hole?
QUESTION [12 upvotes]: The Euler characteristic of a two-dimensional disk is $\chi=1$.
If one blindly interprets the disk as a closed, orientable
surface, then $\chi = 2 - 2g$, and the genus is $g=\frac{1}{2}$.

Is there some way to view a disk as possessing "half a hole" or "half a handle"?

My students asked me and I didn't have a good answer.

REPLY [4 votes]: I know this is an old post but the disk is a once-perforated sphere, and the Euler char formula becomes 2-2g-n, where g is the number of handles and n is the number of perforations.