eventual property

Let $X$ be a set and $P$ a property on the elements of $X$. Let ${(x_i)}_{i\in D}$ be a net ($D$ a directed set) in $X$ (that is, $x_i\in X$). As each $x_i\in X$, $x_i$ either has or does not have property $P$. We say that the net $(x_i)$ has property $P$ above $j\in D$ if $x_i$ has property $P$ for all $i\ge j$. Furthermore, we say that $(x_i)$ eventually has property $P$ if it has property $P$ above some $j\in D$.

Examples.

1. Let $A$ and $B$ be non-empty sets. For $x\in A$, let $P(x)$ be the property that $x\in B$. So $P$ is nothing more than the property of elements being in the intersection of $A$ and $B$. A net ${(x_i)}_{i\in D}$ eventually has $P$ means that for some $j\in D$, the set $\{x_i\mid i\in A\text{, }i\ge j\}\subseteq B$. If $D=\mathbb{Z}$, then we have that $A$ and $B$ eventually coincide.

2. Now, suppose $A$ is a topological space, and $B$ is an open neighborhood of a point $x\in A$. For $y\in A$, let $P_B(y)$ be the property that $y\in B$. Then a net $(x_i)$ has $P_B$ eventually for every neighborhood $B$ of $x$ is a characterization of convergence (to the point $x$, and $x$ is the accumulation point of $(x_i)$).

3. If $A$ is a poset and $B=\{x\}\subseteq A$. For $y\in A$, let $P(y)$ again be the property that $y=x$. Let $(x_i)$ be a net that eventually has property $P$. In other words, $(x_i)$ is eventually constant. In particular, if for every chain $D$, the net ${(x_i)}_{i\in D}$ is eventually constant in $A$, then we have a characterization of the ascending chain condition in $A$.

4. directed net. Let $R$ be a preorder and let ${(x_i)}_{i\in D}$ be a net in $R$. Let $x(D)$ be the image of the net: $x(D)=\{x_i\in R\mid i\in D\}$. Given a fixed $k\in D$ and some $y\in x(D)$, let $P_k(y)$ be the property (on $x(D)$) that $x_k\le y$. Let

   $$S=\{k\in D\mid (x_i)\text{ eventually has }{P}_k\}.$$

   If $S=D$, then we say that the net $(x_i)$ is directed, or that $(x_i)$ is a directed net. In other words, a directed net is a net ${(x_i)}_{i\in D}$ such that for every $i\in D$, there is a $k(i)\in D$, such that $x_i\le x_j$ for all $j\ge k(i)$.

   If ${(x_i)}_{i\in D}$ is a directed net, then $x(D)$ is a directed set: Pick $x_i,x_j\in x(D)$, then there are $k(i),k(j)\in D$ such that $x_i\le x_m$ for all $m\ge k(i)$ and $x_j\le x_n$ for all $n\ge k(j)$. Since $D$ is directed, there is a $t\in D$ such that $t\ge k(i)$ and $t\ge k(j)$. So $x_t\ge x_{k(i)}\ge x_i$ and $x_t\ge x_{k(j)}\ge x_j$.

   However, if ${(x_i)}_{i\in D}$ is a net such that $x(D)$ is directed, $(x_i)$ need not be a directed net. For example, let $D=\{p,q,r\}$ such that $p\le q\le r$, and $R=\{a,b\}$ such that $a\le b$. Define a net $x:D\to R$ 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 $\mathrm{Eventually}(P,X)$ 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