Cantor’s Intersection Theorem

Theorem 1.

Let K1K2K3Kn be a sequence of non-empty, compact subsets of a metric space X. Then the intersection iKi is not empty.


Choose a point xiKi for every i=1,2, Since xiKiK1 is a sequence in a compact set, by Bolzano-Weierstrass TheoremMathworldPlanetmath, there exists a subsequence xij which converges to a point xK1. Notice, however, that for a fixed index n, the sequence xij lies in Kn for all j sufficiently large (namely for all j such that ij>n). So one has xKn. Since this is true for every n, the result follows. ∎

Title Cantor’s Intersection Theorem
Canonical name CantorsIntersectionTheorem
Date of creation 2013-03-22 15:12:35
Last modified on 2013-03-22 15:12:35
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 4
Author paolini (1187)
Entry type Theorem
Classification msc 54E45