Limits of abstract smooth surfaces For $r >0$, let $K_r \subseteq \mathbb{C}$ be the closed subset, $K_r = \mathbb{C} \setminus D(0,r)$. Define $S_r$ to be the quotient of $K_r$ under the identification:
$$ z \sim -z,  \hspace{1cm} z\in \partial D(0,r)$$
Then, $S_r$ is a topological surface, as it is locally homeomorphic to $\mathbb{R}^2$, Hausdorff and second countable. I think that it can also be given the structure of an abstract smooth surface with an appropriate atlas of charts.
Question: is there any meaningful sense in which the limit of $S_r$ can be defined as $r \to 0$?
I was thinking about this as I learnt about on how the Riemann sphere
$\hat{\mathbb{C}}$, functions which are meromorphic over $\mathbb{C}$ can be considered as holomorphic thanks to the addition of the "point at infinity". I am curious about what could happen if instead of adding $\{\infty\}$ to $\mathbb{C}$, we remove $0$ -- resulting in a twice-punctured sphere -- then try to close off (or even glue together) the ends of the "cylinder" to which this twice-punctured sphere is homeomorphic.
 A: Here's a concrete version of the linked post: In polar coordinates $z = |z|e^{i\theta}$, define a mapping from $K_{r}$ to the Cartesian product of the complex plane with itself by
$$
f(z) = (z(1 - r/|z|), rz^{2}/|z|^{2})
= ((\rho-r)e^{i\theta}, re^{2i\theta}).
$$
This mapping is continuous, identifies antipodal points (and no others) on the circle of radius $r$, and the first coordinate maps the interior of $K_{r}$ bijectively (in fact diffeomorphically) to the punctured plane. Bijectivity in the first coordinate corresponds to the surprising fact that the Möbius-band, despite its twist, collapses to a disk when the central circle is collapsed. In other words, a Möbius band without its central circle is diffeomorphic to a punctured disk.
To see what happens in the annulus $\{r \leq |z| \leq R\}$, we may glue two copies of the preceding picture (or write down an appropriate mapping), yielding a Klein bottle (connected sum of two Möbious bands, a.k.a., cross caps) that collapses to a sphere (connected sum of two disks) as $r \to 0$ and $R \to \infty$.
