Bounded sets and diameters lead to nested closed sets in complete metric spaces. A decreasing sequence of nonempty closed sets with diameters tending to zero should determine one point, but this conclusion depends on completeness. The Cantor intersection theorem captures this principle and explains why closedness, shrinking diameter, and completeness all matter. :::definition[Nested Sequence of Sets] Let $\{F_n\}$ be a sequence of subsets of a set $X$. The sequence is $nested$ if $F_{n+1}\subset F_n$ for every $n\in\mathbb{N}$, so $F_1\supset F_2\supset F_3\supset\cdots$. ::: :::example[Nested Open Intervals] In $\mathbb{R}$, the intervals $I_n=(0,1/n)$ are nested, but $\bigcap_{n=1}^{\infty}I_n=\varnothing$. Thus openness is not enough. ::: :::example[Nested Closed Unbounded Intervals] The sets $F_n=[n,\infty)$ are nested closed subsets of $\mathbb{R}$, but their intersection is empty. Thus closedness alone is not enough. ::: :::example[Nested Sets in the Rationals] In $\mathbb{Q}$, sets that shrink toward $\sqrt{2}$ may be closed in $\mathbb{Q}$ with diameters tending to zero but still have empty intersection, because $\sqrt{2}\notin\mathbb{Q}$. ::: :::theorem[Cantor Intersection Theorem] Let $(X,d)$ be a metric space. Then $X$ is complete if and only if every nested sequence $\{F_n\}$ of nonempty closed subsets of $X$ satisfying $\operatorname{diam}(F_n)\to0$ has intersection consisting of exactly one point. ::: :::proof Given that $(X,d)$ is a metric space. To prove the Cantor intersection theorem. Suppose $X$ is complete. Choose $x_n\in F_n$. Since $\operatorname{diam}(F_n)\to0$, for every $\varepsilon>0$ there exists $N$ such that $\operatorname{diam}(F_N)<\varepsilon$. If $m,n\geq N$, then $x_m,x_n\in F_N$, so $d(x_m,x_n)<\varepsilon$. Thus $\{x_n\}$ is Cauchy and converges to some $x\in X$. Since each $F_N$ contains all tails of the sequence and is closed, $x\in F_N$ for every $N$. Hence the intersection is nonempty. If $x,y$ both lie in the intersection, then $d(x,y)\leq\operatorname{diam}(F_n)$ for every $n$. Letting $n\to\infty$ gives $d(x,y)=0$, so $x=y$. Conversely, let $\{x_n\}$ be a Cauchy sequence in $X$ and set $F_n=\overline{\{x_k:k\geq n\}}$. These sets are nonempty, closed, nested, and have diameters tending to zero. By the stated property, their intersection contains one point $x$. The Cauchy tail construction gives $x_n\to x$. Hence $X$ is complete. $\square$ ::: :::solved-problem Let $X=\{(x_1,x_2)\in\mathbb{R}^2:|x_1|<2,\ |x_2|<1\}$. Show that for any $x\in X$ and $r\geq2\sqrt{5}$, $S_X(x,r)=X$. ::: :::solution For any $x,y\in X$, $|x_1-y_1|<4$ and $|x_2-y_2|<2$. Hence $d(x,y)<\sqrt{4^2+2^2}=2\sqrt{5}$. Therefore every $y\in X$ lies in $S_X(x,r)$ whenever $r\geq2\sqrt{5}$. $\square$ ::: :::exercise [1] State the main definition from this lesson. [2] State the main theorem from this lesson. [3] Give one example illustrating the theorem. [4] Explain one common mistake related to this topic. ::: :::answer [1] The main definition is the first titled definition in the lesson. [2] The main theorem is the titled theorem proved in the lesson. [3] The examples in the lesson provide standard choices. [4] A common mistake is to ignore the metric or the ambient space when applying the definition. ::: :::faq[Frequently Asked Questions] Q: Why is the ambient metric space important? A: Openness, closedness, closure, and boundary are interpreted relative to the chosen metric space. Q: Can the same set behave differently in different metric spaces? A: Yes. Changing the metric or the ambient space may change topological behaviour. Q: How should students check a definition? A: They should return to balls, closures, and sequences, depending on the context. ::: :::call-to-action[Continue from Nested Closed Sets in Complete Spaces] subtitle: Review the definitions and proofs before moving through the metric topology sequence. button: Next Lesson | /functional-analysis/metric-spaces/topological-structure-of-metric-spaces/relative-topology-of-subspaces button-ghost: Previous Lesson | /functional-analysis/metric-spaces/topological-structure-of-metric-spaces/bounded-sets-and-diameters :::