nLab
subnet

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Analysis

Contents

Idea

The definition of the concept of sub-nets of a net requires some care. The point of the definition is to ensure that compact spaces are equivalently those for which every net has a converging subnet.

Definition

There are several different definitions of ‘subnet’ in the literature, all of which intend to generalise the concept of subsequences. We state them now in order of increasing generality. Note that it is Definition 3 which is correct in that it corresponds precisely to refinement of filters. However, the other two definitions (def. 1, def. 2) are sufficient (in a sense made precise by theorem 1 below) and may be easier to work with.

Definition

(Willard, 1970).

Given a net (x α)(x_{\alpha}) with index set AA, and a net (y β)(y_{\beta}) with an index set BB, we say that yy is a subnet of xx if:

We have a function f:BAf\colon B \to A such that

  • ff maps xx to yy (that is, for every βB\beta \in B, y β=x f(β)y_{\beta} = x_{f(\beta)});
  • ff is monotone (that is, for every β 1β 2B\beta_1 \geq \beta_2 \in B, f(β 1)f(β 2)f(\beta_1) \geq f(\beta_2));
  • ff is cofinal (that is, for every αA\alpha \in A there is a βB\beta \in B such that f(β)αf(\beta) \geq \alpha).
Definition

(Kelley, 1955).

Given a net (x α)(x_{\alpha}) with index set AA, and a net (y β)(y_{\beta}) with an index set BB, we say that yy is a subnet of xx if:

We have a function f:BAf\colon B \to A such that

  • ff maps xx to yy (that is, for every βB\beta \in B, y β=x f(β)y_{\beta} = x_{f(\beta)});
  • ff is strongly cofinal (that is, for every αA\alpha \in A there is a βB\beta \in B such that, for every β 1βB\beta_1 \geq \beta \in B, f(β 1)αf(\beta_1) \geq \alpha).
Remark

Notice that the function ff in definitions 1 and 2 is not required to be an injection, and it need not be. As a result, a sequence regarded as a net in general has more sub-nets than it has sub-sequences.

Definition

(Smiley, 1957; Årnes & Andenæs, 1972).

Given a net (x α)(x_{\alpha}) with index set AA, and a net (y β)(y_{\beta}) with an index set BB, we say that yy is a subnet of xx if:

The eventuality filter of yy refines the eventuality filter of xx. (Explicitly, for every αA\alpha \in A there is a βB\beta \in B such that, for every β 1βB\beta_1 \geq \beta \in B there is an α 1αA\alpha_1 \geq \alpha \in A such that y β 1=x α 1y_{\beta_1} = x_{\alpha_1}.)

The equivalence between these definitions is as follows:

Theorem

(Schechter, 1996).

  1. If yy is a (1)-subnet of xx, then yy is also a (2)-subnet of xx, using the same function ff.
  2. If yy is a (2)-subnet of xx, then yy is also a (3)-subnet of xx.
  3. If yy is a (3)-subnet of xx, then there is some net zz such that
    • zz is equivalent to yy in the sense that yy and zz are (3)-subnets of each other, and
    • zz is a (1)-subnet of xx, using some function.

So from the perspective of definition (3), there are enough (1)-subnets and (2)-subnets, up to equivalence.

References

A textbook account is in

Lecture notes include

Created on April 23, 2017 09:11:29 by Urs Schreiber (92.218.150.85)