# 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 $i\geq j$. Furthermore, we say that $(x_{i})$ eventually has property $P$ if it has property $P$ above some $j\in D$.

Examples.

1. 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 $\{x_{i}\mid i\in A\mbox{, }i\geq j\}\subseteq B$. If $D=\mathbb{Z}$, then we have that $A$ and $B$ eventually coincide.

2. 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. 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. 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 $x_{k}\leq y$. Let

 $S=\{k\in D\mid(x_{i})\mbox{ eventually has }P_{k}\}.$

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 $x_{i}\leq x_{j}$ for all $j\geq k(i)$.

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 $x_{i}\leq x_{m}$ for all $m\geq k(i)$ and $x_{j}\leq x_{n}$ for all $n\geq k(j)$. Since $D$ is directed, there is a $t\in D$ such that $t\geq k(i)$ and $t\geq k(j)$. So $x_{t}\geq x_{k(i)}\geq x_{i}$ and $x_{t}\geq x_{k(j)}\geq x_{j}$.

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 $p\leq q\leq r$, and $R=\{a,b\}$ such that $a\leq b$. 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 $\operatorname{Eventually}(P,X)$ to denote its dependence on $X$ and $P$.

Title eventual property EventualProperty 2013-03-22 16:34:45 2013-03-22 16:34:45 CWoo (3771) CWoo (3771) 16 CWoo (3771) Definition msc 06A06 residually constant eventually directed net eventually constant