= Total Gaussian curvature of a punctured surface with a singular compactification point
The topology and compactness of a one-point closure do not determine the total curvature when the added point need not be a regular point of the surface. Rotate a smooth simple arc in the half-plane about its boundary axis. Let its arc-length parameter be $s\in[0,L)$, let its radius be $r(s)>0$ for $0<s<L$, and arrange
$$
r(0)=0,\quad r'(0)=1,\qquad
r(s)\to0,\quad z(s)\to z_L,\quad r'(s)\to-a
$$
as $s\uparrow L$, where $0\leq a<1$. The initial end closes smoothly, while the terminal point is omitted. The resulting surface is diffeomorphic to a sphere minus one point and has compact one-point closure. Since $K=-r''/r$ and $dA=r\,ds\,d\theta$,
$$
\int_SK\,dA=-2\pi[r']_0^L=2\pi(1+a),
$$
which need not equal $4\pi$.
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