Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 2G (Topics in Analysis)

The Baire category theorem says that a complete metric space cannot be a countable union of nowhere-dense closed sets. Equivalently, every countable intersection of open dense sets is dense.

A point $x$ of a metric space is an isolated point when some $\epsilon >0$ satisfies $B(x,\epsilon )=\{x\}$.

False. The rational numbers with the Euclidean metric are countable and have no isolated point.

False. Every discrete metric space is complete and every one of its points is isolated; a singleton is the smallest example.

False. The interval $(0,1)$ with the Euclidean metric is uncountable and has no isolated points, but the Cauchy sequence $1/n$ has no limit in the space.

False. A singleton, or $\mathbb{Z}$ with the discrete metric, is complete and countable.

True. If a nonempty countable complete metric space had no isolated point, every singleton would be closed and nowhere dense. Their countable union would be the whole space, contradicting the Baire category theorem. This is the basic isolated points in a countable complete metric space argument.

False. The subspaceis closed and hence complete, and it is countable; however, $0$ is not isolated.

True. Part (v) supplies one isolated point $p$. Since $\{p\}$ is open, $X\setminus \{p\}$ is closed and therefore complete. It is nonempty when $X$ has at least two points, so part (v) applied to this subspace supplies an isolated point $q$. Because $d(p,q)>0$, a sufficiently small ball showing that $q$ is isolated in the subspace also avoids $p$; hence $q$ is isolated in $X$.

True. If there were only finitely many isolated points, delete all of them. The remainder would be a nonempty closed, countable, complete subspace, so part (v) would give it an isolated point. Its positive distance from the finite deleted set would make it isolated in the original space as well, a contradiction. Thus the stronger isolated points in a countable complete metric space result applies.