Exercise 2.10

We see that $d(x,x)=0$, and $d(x,y)\ne 0$ whenever $x\ne y$. We also have $d(x,y)=d(y,x)$. Lastly, consider

The inequality is obvious if $x=z$, and otherwise either $y\ne x$ or $y\ne z$, so that the inequality still holds. The function $d$ is thus a metric.

Note that every point set is open, since $\{x\}=N_{1∕2}(x)$. Consequently, every subset of $X$ is open, and therefore every subset of $X$ is closed. Let $S$ be a subset of $X$, and consider the open cover ${\{x\}}_{x\in S}$. This cover has a finite subcover iff $S$ is finite. Any compact set must therefore be finite, and conversely, it is easy to see that any finite set is compact. It is thus clear that the compact sets are precisely those that are finite.