Cantor's intersection theorem

Cantor's intersection theorem states that nested nonempty closed sets in a compact space have nonempty intersection. In a complete metric space, nested nonempty closed sets whose diameters tend to zero intersect in exactly one point.

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Nested set: A sequence of sets is nested when each set is contained in the preceding set. The theorem’s sequence must keep narrowing by inclusion.
Closed set: A set is closed when its complement is open, equivalently when it contains all its limit points. Closedness is required for the nested sets in both standard forms.
Bisection method: The bisection method repeatedly halves an interval to approximate a root or solve an equation. Its nested closed intervals have diameters tending to zero, guaranteeing a unique limiting point.
Cantor intersection theorem for compact sets: A compact-set formulation states that a decreasing sequence of nonempty compact sets has nonempty intersection. Unlike the closed-set version, compactness is imposed on every set in the sequence.
Compact space: A topological space is compact if every open cover has a finite subcover. Compactness ensures nested nonempty closed sets cannot lose all their common points.
Intersection of sets: The intersection of sets consists of elements belonging to every set in the family. The theorem guarantees that this common part is not empty.
Nested intervals theorem: The nested intervals theorem states that nested nonempty closed bounded intervals in the real line intersect. It is the real-line form used directly to pin down limits and real numbers.
Cantor intersection theorem for complete metric spaces: A metric-space formulation says nested nonempty closed sets with diameters tending to zero meet in one point when the space is complete. This version adds a metric and shrinking diameters to obtain uniqueness.
Complete metric space: A metric space is complete if every Cauchy sequence converges to a point in that space. Completeness supplies the limit point in the metric-space version.
Bolzano–Weierstrass theorem: Every bounded sequence in Euclidean space has a convergent subsequence. Its subsequence compactness principle underlies familiar metric-space proofs.