Euler characteristic 1: Half a hole? 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.
 A: 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. 
A: The connected sum of two disks is an annulus. If you think of an annulus as being a hole, then I suppose a disk is half a hole. 
A: Trying to use $\chi = 2 - 2g$ to describe things that aren't closed orientable surfaces is missing the point, I think. In my opinion one should think of the Euler characteristic of a compact space as a homotopy-invariant refinement of the cardinality of a finite set; see this blog post. A closed disk is contractible, so has Euler characteristic $1$, and that's the most transparent interpretation of it. You might also be interested in the argument in the blog post that derives $\chi = 2 - 2g$ from homotopy-invariance and inclusion-exclusion. 
The thing that possesses "half a hole" isn't the closed disk; if anything, it's $\mathbb{R}P^2$, which also has Euler characteristic $1$. And this is totally sensible as it can be described as the quotient of $S^2$ by an action of $\mathbb{Z}/2\mathbb{Z}$. 
