I'm totally stuck with the following question, I even don't know how to start:
For each natural number $n$ we consider a space $X_n$ that is obtained by removing $n$ distinct points from $\mathbb{R}^2$. We consider the 1-point compactification $X_n^+$ and we denote by $\infty_n\in X_n^+$ the point at infinity (so that $X_n^+=X_n\cup \{\infty_n\}$). Show that
- $X_n^+$ can be embedded in $\mathbb{R}^3$. I do not have to write down explicit formulas for the embedding, but I have to explain my reasoning using pictures and mention what result(s) I use in order to reach my final conclusion
- If $X_n$ and $X_m$ are homeomorphic, then $n=m$.
Both questions I find very hard to answer. I really hope someone can help me!
HINT: Start by showing that $X_0^+$ is homeomorphic to $S^2$. Then convince yourself that $X_1$ is homeomorphic to the cylinder $S^1\times\Bbb R$. If you added two points at infinity instead of one, one at each end of the cylinder, you would compactify it into $S^2$ by in essence closing off the ends; if you now identify the two new points, you are in effect starting with $S^2$ and identifying the north and south poles. The result is a torus whose inner radius has been shrunk to $0$, so that instead of a hole in the centre it has a single point. Even though we did this in two steps, first adding two points and then identifying them, it turns out that we really have got $X_1^+$.
To extend the basic idea, try to convince yourself that $X_n$ is homeomorphic to the space obtained by removing $n+1$ points from $S^2$; $X_n^+$ is then obtained by filling all $n+1$ holes with a single point.