eventual property

Let X be a set and P a property on the elements of X. Let (xi)iD be a net (D a directed setMathworldPlanetmath) in X (that is, xiX). As each xiX, xi either has or does not have property P. We say that the net (xi) has property P above jD if xi has property P for all ij. Furthermore, we say that (xi) eventually has property P if it has property P above some jD.


  1. 1.

    Let A and B be non-empty sets. For xA, let P(x) be the property that xB. So P is nothing more than the property of elements being in the intersectionMathworldPlanetmathPlanetmath of A and B. A net (xi)iD eventually has P means that for some jD, the set {xiiAij}B. If D=, then we have that A and B eventually coincide.

  2. 2.

    Now, suppose A is a topological spaceMathworldPlanetmath, and B is an open neighborhood of a point xA. For yA, let PB(y) be the property that yB. Then a net (xi) has PB eventually for every neighborhood B of x is a characterization of convergence (to the point x, and x is the accumulation pointPlanetmathPlanetmath of (xi)).

  3. 3.

    If A is a poset and B={x}A. For yA, let P(y) again be the property that y=x. Let (xi) be a net that eventually has property P. In other words, (xi) is eventually constant. In particular, if for every chain D, the net (xi)iD is eventually constant in A, then we have a characterization of the ascending chain conditionMathworldPlanetmathPlanetmathPlanetmath in A.

  4. 4.

    directed net. Let R be a preorderMathworldPlanetmath and let (xi)iD be a net in R. Let x(D) be the image of the net: x(D)={xiRiD}. Given a fixed kD and some yx(D), let Pk(y) be the property (on x(D)) that xky. Let

    S={kD(xi) eventually has Pk}.

    If S=D, then we say that the net (xi) is directed, or that (xi) is a directed net. In other words, a directed net is a net (xi)iD such that for every iD, there is a k(i)D, such that xixj for all jk(i).

    If (xi)iD is a directed net, then x(D) is a directed set: Pick xi,xjx(D), then there are k(i),k(j)D such that xixm for all mk(i) and xjxn for all nk(j). Since D is directed, there is a tD such that tk(i) and tk(j). So xtxk(i)xi and xtxk(j)xj.

    However, if (xi)iD is a net such that x(D) is directed, (xi) need not be a directed net. For example, let D={p,q,r} such that pqr, and R={a,b} such that ab. Define a net x:DR by x(p)=x(r)=b and x(q)=a. Then x is not a directed net.

Remark. The eventual property is a property on the class of nets (on a given set X and a given property P). We can write Eventually(P,X) to denote its dependence on X and P.

Title eventual property
Canonical name EventualProperty
Date of creation 2013-03-22 16:34:45
Last modified on 2013-03-22 16:34:45
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 16
Author CWoo (3771)
Entry type Definition
Classification msc 06A06
Synonym residually constant
Defines eventually
Defines directed net
Defines eventually constant