Cantor's intersection theorem

Statement

Let $X$ be a topological space. Let $(C_n)_{n \ge 1}$ be a sequence of nonempty compact closed subsets of $X$ with $C_1 \supseteq C_2 \supseteq C_3 \supseteq \cdots$. Then the intersection $\bigcap_{n \ge 1} C_n$ is not empty.

Remarks

The sets must be compact: in $\mathbb{R}$, the closed sets $[n, \infty)$ are nested and nonempty, but their intersection is empty. For $X = \mathbb{R}$, the sets can be nested closed bounded intervals $[a_n, b_n]$. There is a variant for metric spaces. A metric space is complete if and only if each nested sequence of nonempty closed sets whose diameters tend to $0$ has nonempty intersection. The intersection is then one point.

Explanations

L1 human

If you have a chain of nonempty compact sets, each inside the one before, at least one point is in all of them. For example, the intervals $[0, 1/n]$ are nested, and $0$ is in all of them.