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.
Consider the map$$\begin{array}{rccc}s\colon&\mathbb R&\longrightarrow&S^1\\&x&\mapsto&\left(\dfrac{1-x^2}{1+x^2},\dfrac{2x}{1+x^2}\right).\end{array}$$Then $s$ is a homeomorphism between $\mathbb R$ and $S^1\setminus\bigl\{(-1,0)\bigr\}$. So, since $S^1\setminus\bigl\{(-1,0)\bigr\}$ is dense in $S^1$ and $S^1\setminus\left(S^1\setminus\bigl\{(-1,0)\bigr\}\right)$ consists of a single point, $S^1$ is the one-point compactification of $\mathbb R$. More generally, if $\theta\in\mathbb R$, then$$\begin{array}{rccc}s_\theta\colon&\mathbb R&\longrightarrow&S^1\\&x&\mapsto&\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}.\left(\dfrac{1-x^2}{1+x^2},\dfrac{2x}{1+x^2}\right)\end{array}$$is a homeomorphism between $\mathbb R$ and $S^1\setminus\bigl\{(-\cos\theta,-\sin\theta)\bigr\}$.
A similar argument applies to $\mathbb R^2$ and $S^2$. Just consider the map:$$\begin{array}{rccc}\psi\colon&\mathbb R^2&\longrightarrow&S^2\\&(x,y) &\mapsto&\left(\frac{2x}{x^2+y^2+1},\frac{2y}{x^2+y^2+1},\frac{x^2+y^2-1}{x^2+y^2+1}\right).\end{array}$$
It is a homeomorphism between $\mathbb R^2$ and $S^2\setminus\{(0,0,1)\}$.
Best Answer
It means:
Munkres then gives a construction that explains how to take $X$ and construct the corresponding $Y$. And because $Y$ is uniquely determined, that one construction is the whole story; there are no other ways to produce a one-point compactification of a space.
Munkres’ presentation of this is a little hard to follow. What is the one-point compactification of $\mathbb{Z}_{+}$? discusses it and may help a little bit.