every net has a universal subnet


Theorem - (Kelley’s theorem) - Let X be a non-empty set. Every net (xα)α∈𝒜 in X has a universalPlanetmathPlanetmath subnet (http://planetmath.org/Ultranet). That is, there is a subnet such that for every E⊆X either the subnet is eventually in E or eventually in X-E.

Proof : Let ℱ be a section filter for the net (xα)α∈𝒜.

Let 𝒟={(α,U):α∈𝒜,U∈ℱ,xα∈U}. 𝒟 is a directed setMathworldPlanetmath under the order relation given by

(α,U)≤(β,V)⟺{α≤βV⊆U

The map f:𝒟⟶𝒜 defined by f⁢(α,U):=α is order preserving and cofinal. Therefore there is a subnet (y(α,U))(α,U)∈𝒟 of (xα)α∈𝒜 associated with the map f (that is, y(α,U)=xα).

We now prove that (y(α,U))(α,U)∈𝒟 is a net.

Let E⊆X. We have that (y(α,U))(α,U)∈𝒟 is frequently in E or frequently in X-E.

Suppose (y(α,U))(α,U)∈𝒟 is frequently in E.

Let A∈ℱ and S⁢(α):={xβ:α≤β}. We have that S⁢(α)∈ℱ by definition of section filter.

As ℱ is a filter, A∩S⁢(α)≠∅ and so there exists β with α≤β such that xβ∈A. Hence, (β,A)∈𝒟.

As (y(α,U))(α,U)∈𝒟 is frequently in E, there exists (γ,B)∈𝒟 with (β,A)≤(γ,B) such that y(γ,B)∈E.

Also, y(γ,B) is in B, and therefore, in A. So A∩E≠∅.

We conclude that E∩A≠∅ for every A∈ℱ. Therefore, ℱ∪{E} a filter in X. As ℱ is a maximal filter we conclude that E∈ℱ, and consequently, (γ,E)∈𝒟.

We can now see that for every (δ,C) with (γ,E)≤(δ,C), y(δ,C) is in C and so is in E. Therefore, (y(α,U))(α,U)∈𝒟 is eventually in E.

Remark: If (y(α,U))(α,U)∈𝒟 is frequently in X-E, by an analogous we can conclude that it is eventually in X-E.

This proves that (y(α,U))(α,U)∈𝒟 is a subnet of (xα)α∈𝒜. □

Title every net has a universal subnet
Canonical name EveryNetHasAUniversalSubnet
Date of creation 2013-03-22 17:25:16
Last modified on 2013-03-22 17:25:16
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 8
Author asteroid (17536)
Entry type Theorem
Classification msc 54A20
Synonym Kelley’s theorem
Related topic Ultranet