Why does a once punctured sphere have no boundary? Why does a once punctured sphere have no boundary? Also, why can we think of a once punctured sphere just as $\mathbb{R}^2$? Is it because we can project it onto $\mathbb{R}^2$?
 A: In general an open subset of a manifold without boundary is a manifold without boundary.  The punctured sphere is an open subset of the sphere since we have removed a closed set (consisting of a single point).  
The punctured sphere is mapped homeomorphically onto $\mathbb{R}^2$ via Steriographic projection.  http://en.wikipedia.org/wiki/Stereographic_projection
A: The boundary of a bidimensional manifold is the set of points that has a neighbourhood homemorphic with the neighbourhood of a point that belongs to the boundary of a closed half-plane, and in a punctured sphere there are no such points.
Alternatively, the boundary of a boundary is always $\emptyset$. And the punctured sphere is homeomorphic to $\Bbb R^2$, which is the boundary of a half-space.
A: [Most of this answer is copied from here].
To understand the way that $R^2$ is homeomorphic to the punctured sphere, it is helpful to first consider the situation in one dimension.  Just as $R^2$ is homeomorphic to $S^2$ with a point removed, so is the $R^1$ is homeomorphic to $S^1$ with a point removed.
The idea is this: A circle $\bigl(S^1\bigr)$ is like a line  $\bigl(R^1\bigr)$ with the ends glued together; or conversely a line is like a circle with one point deleted.  There is an easy homeomorphism:  Consider the circle with center at $\left(0, \frac1 2\right)$.  For each point $P$ on the circle, draw a line through $P$ and $N = (0, 1)$. 

This line intersects the $x$-axis in one point $P'$, and this is a homeomorphism between the $x$-axis and the circle minus the point $N$ itself.
One can do a similar mapping for the sphere onto a plane: for each point $P$ on the sphere, draw a line from the north pole $N$ of the sphere through $P$ to finds its intersection $P'$ on the plane. This is a one-to-one correspondence between the points of the plane and the points of the sphere other than the north pole.
The mapping  is called a stereographic projection.

Again the idea is that if you take a sphere and delete a point, you can stretch out the part around the deleted point and take it out to infinity, flatten it out, and what you get is the plane.  Or you can do the same thing in reverse, adding a "point at infinity" that brings all the far-away parts of the plane together; this "point at infinity" is the "one point" in the "one-point compactification".
The construction in higher dimensions is completely analogous.
