I have a few questions from the following paragraph of Topology by James Munkres page 377.
Let $C$ be a compact subspace of $S^2$ and $b\in S^2-C$.
The paragraph:
The stereographic projection $h: S^2-b \to \mathbb{R}^2$ can be extended
to a homeomorphism $H$ of $S^2$ with one-point compactification $\mathbb{R}^2\cup\{\infty\}$ of $\mathbb{R}^2$, merely by setting $H(b)=\infty$. If $U_{\beta}$ is the component of $S^2-C$ containing $b$,
then $H(U_{\beta})$ is a neighborhood of $\infty$ in $\mathbb{R}^2\cup\infty$. Therefore, $V_{\beta} = h(U_{\beta} – b)$ is unbounded; since its complement $\mathbb{R}^2-V_{\beta}$ is compact, all other components of $\mathbb{R}^2-h(C)$ are bounded.
My Questions:
1) By $V_{\beta}$ being unbounded, does this simply mean that it does not contain the point $\infty$? From this, can we conclude that any subset
of the compactification of $\mathbb{R}^2$ not including $\infty$ is bounded?
2) How different are the topologies of $\mathbb{R}^2$ and $\mathbb{R}^2\cup\{\infty\}$?
3) I am not sure why the complement $\mathbb{R}^2-V_{\beta}$ is compact, and it seems to me that we are using Heine-Borel to conclude the result right after that which is weird to me since we are working the compactification of $\mathbb{R}^2$ and not $\mathbb{R}^2$ itself (iirc, Heine-Borel only holds for the standard topology).
4) (Bonus) Help me rename this question to something… better.
Best Answer
1) "Unbounded" is a property of subsets of $\mathbb{R}^2$, not the one-point compactification. Note that $V_\beta$, the set described as unbounded, is obtained by applying $h$, which only takes values in $\mathbb{R}^2$. So unbounded means what it usually does: not contained in a ball.
2) Not very. When we take the one-point compactification, all of the sets that were open in $\mathbb{R}^2$ stay open. The only open sets that get added are the neighborhoods of $\infty$: complements of sets that were compact in $\mathbb{R}^2$.
In particular, if you look at $\mathbb{R}^2$ as a subspace of $\mathbb{R}^2\cup\{\infty\}$ with the subspace topology, you'll get back the standard topology on $\mathbb{R}^2$ we started with, since we didn't mess around with the open sets that don't contain the new point $\infty$.
3) If you're willing to assume what's been said previously in the paragraph (that $H$ is a homeomorphism and that $H(U_\beta)$ is open--it's worth working these out if you're not sure why), then since $H(U_\beta)$ is a neighborhood of $\infty$, it's characterized by the definition of the one-point compactification: it's the complement of a compact set in $\mathbb{R}^2$. If you chase the definitions of things, you can see that this compact set, $\mathbb{R}^2 \cup \{\infty\} - H(U_\beta)$, is the same as $\mathbb{R}^2 - V_\beta$.
As for the application of the Heine-Borel theorem immediately afterwards, note again that we're working with $\mathbb{R}^2 - V_\beta$ and $\mathbb{R}^2 - h(C)$, which are subsets of $\mathbb{R}^2$, not its one-point compactification. You are correct that Heine-Borel only works on subsets of $\mathbb{R}^n$.
4) You might want the name to mention that you're working with one-point compactification.