universal nets in compact spaces are convergent

Theorem - A universal net (xα)α𝒜 in a compact space X is convergent.

Proof : Suppose by contradictionMathworldPlanetmathPlanetmath that (xα)α𝒜 was not convergent. Then for every xX we would find neighborhoodsMathworldPlanetmathPlanetmath Ux such that


The collectionMathworldPlanetmath of all this neighborhoods cover X, and as X is compactPlanetmathPlanetmath, a finite number Ux1,Ux2,,Uxn also cover X.

The net (xα)α𝒜 is not eventually in Uxk so it must be eventually in X-Uxk (because it is a net). Therefore we can find αk𝒜 such that


Because we have a finite number α1,α2,αn𝒜 we can find γ𝒜 such that αkγ for each 1kn.

Then xγX-Uxk for all k, i.e. xγUxk for all k. But Ux1,Ux2,,Uxn cover X and thus we have a contradiction.

Title universal nets in compact spaces are convergent
Canonical name UniversalNetsInCompactSpacesAreConvergent
Date of creation 2013-03-22 17:31:29
Last modified on 2013-03-22 17:31:29
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 4
Author asteroid (17536)
Entry type Theorem
Classification msc 54A20