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 $P$.