topology via converging nets

Given a topological spaceMathworldPlanetmath X, one can define the concept of convergence of a sequence, and more generally, the convergence of a net. Conversely, given a set X, a class of nets, and a suitable definition of “convergence” of a net, we can topologize X. The procedure is done as follows:

Let C be the class of all pairs of the form (x,y) where x is a net in X and y is an element of X. For any subset U of X with y∈U, we say that a net x convergesPlanetmathPlanetmath to y with respect to U if x is eventually in U. We denote this by x→Uy. Let

𝒯:={U⊆X∣(x,y)∈C⁢ and ⁢y∈U⁢ imply ⁢x→Uy}.

Then 𝒯 is a topology on X.

Clearly x→Xy for any pair (x,y)∈C. In addition, x→∅y is vacuously true. For any U,V∈𝒯, we want to show that W:=U∩V∈𝒯. Since x is eventually in U and V, there are i,j∈D (where D is the domain of x), such that xr∈U and xs∈V for all r≥i and s≥j. Since D is directed, there is a k∈D such that k≥i and k≥j. It is clear that xk∈W and that any t≥k we have that xt∈W as well. Next, if Uα are sets in 𝒯, we want to show their union U:=⋃{Uα} is also in 𝒯. If y is a point in U then y is a point in some Uα. Since (x,y)∈C with x is eventually in Uα, we have that x is eventually in U as well. ∎

Remark. The above can be generalized. In fact, if the class of pairs (x,y) satisfies some “axioms” that are commonly found as properties of convergence, then X can be topologized. Specifically, let X be a set and C again be the class of all pairs (x,y) as described above. A subclass 𝒞 of C is called a convergence class if the following conditions are satisfied

  1. 1.

    x is a constant net with value y∈X, then (x,y)∈𝒞

  2. 2.

    (x,y)∈𝒞 implies (z,y)∈𝒞 for any subnet z of x

  3. 3.

    if every subnet z of a net x has a subnet t with (t,y)∈𝒞, then (x,y)∈𝒞

  4. 4.

    suppose (x,y)∈𝒞 with D=dom⁡(x), and for each i∈D, we have that (zi,xi)∈𝒞, with Di=dom⁡(zi). Then (z,x)∈𝒞, where z is the net whose domain is D×F with F:=∏{Di∣i∈D}, given by z⁢(i,f)=(i,f⁢(i)).

If (x,y)∈𝒞, we write x→y or limD⁡x=y. The last condition can then be visualized as


which is reminiscent of Cantor’s diagonal argument.

Now, for any subset A of X, we define Ac to be the subset of X consisting of all points y∈X such that there is a net x in A with x→y. It can be shown that c is a closure operatorPlanetmathPlanetmathPlanetmath, which induces a topology 𝒯𝒞 on X. Furthermore, under this induced topology, the notion of converging nets (as defined by the topology) is exactly the same as the notion of convergence described by the convergence class 𝒞.

In addition, it may be shown that there is a one-to-one correspondence between the topologies and the convergence classes on the set X. The correspondence is order reversing in the sense that if 𝒞1⊆𝒞2 as convergent classes, then 𝒯𝒞2⊆𝒯𝒞1 as topologies.

Title topology via converging nets
Canonical name TopologyViaConvergingNets
Date of creation 2013-03-22 17:14:27
Last modified on 2013-03-22 17:14:27
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 54A20