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 $\lbrace x_i\mid i\in A\mbox{, } i\ge j \rbrace \subseteq B$ . 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 $B=\lbrace x\rbrace \subseteq A$ . 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: $x(D)=\lbrace x_i\in R \mid i\in D\rbrace$ . 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 $$S=\lbrace k \in D \mid (x_i)\mbox{ eventually has }P_k\rbrace.$$ 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 $D=\lbrace p,q,r\rbrace$ such that $p\le q\le r$ , and $R=\lbrace a,b\rbrace$ 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 $\operatorname{Eventually}(P,X)$ to denote its dependence on $X$ and $P$ .