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 nonempty 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  20130322 16:34:45 
Last modified on  20130322 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 