I think you are reading the wrong part of your book. That definition (top of p. 185 in my edition) is immediately preceded by a Theorem 29.1, which is a long explanation of how one can take a noncompact locally-compact Hausdorff space $X$ and embed it into a compact space $\def\Y{X^\ast}\Y$ that has exactly one point more than $X$. This larger space $\Y$ is the one-point compactification of a space $X$, sometimes called the Alexandroff compactification of $X$.
The one-point compactification
$\Y$ consists of $X\cup \{\infty\}$, where $\infty$ is some new point that is not an element of $X$. The topology is as follows:
- If $G$ is an open subset of $X$, then $G$ is also an open set of $\Y$
- If $C$ is a compact subset of $X$, then $\{\infty\}\cup(X\setminus C)$ is an open set of $\Y$
Theorem 29.1 shows that if $X$ is a Hausdorff space that is locally compact but not compact, then $\Y$, with the topology described above, is a compact Hausdorff space of which $X$ is a subspace. Theorem 29.1 also shows that any one-point compactification of $X$ must be homeomorphic to the $\Y$ described above, so that the one-point compactification of $X$ is essentially unique. This is the construction Munkres wants you to consider.
This construction is the topological formalization of the idea of taking an infinite space $X$ and "adding a point at infinity". We do this, for example, with the complex numbers, to obtain the Riemann sphere. (Ignore this example if you don't know about the Riemann sphere.) A simpler example is that the one-point compactification of $\Bbb R$ is (homeomorphic to) $S^1$, the circle: the two ends at infinity are brought together and joined at the new point $\infty$.
I hope the question makes more sense in this light.
A word of advice: In some subjects, and on standardized tests, you can read the question first, then go back and skim the material looking for something pertinent, and then answer the question without reading all the material. In advanced mathematics, this strategy will not work. You have to adopt a different strategy. First read over the entire chapter, very slowly, taking time to understand and digest each sentence before you move on to the next one. This may take several days, or more. Then do the exercises.
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.
Best Answer
Stereographic projection: the north pole $P$ corresponds to the point at infinity, the South pole to zero.
In Cartesian coordinates it's $(x,y,z)\to (\frac y{1-x},\frac z{1-x})$.
As for mapping two arbitrary points on the sphere to the origin and infinity, see the comment by @Cheerful Parsnip.