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Version 6 |
| 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 $P$ \emph{above} $j\in D$ if $x_i$ has property $P$ for all $i\ge j$. Furthermore, we say that $(x_i)$ \emph{eventually} has property $P$ if it has property $P$ above some $j\in D$. |
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 $P$ \emph{above} $j\in D$ if $x_i$ has property $P$ for all $i\ge j$. Furthermore, we say that $(x_i)$ \emph{eventually} has property $P$ if it has property $P$ above some $j\in D$. |
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| \textbf{Examples}. |
For example, let $A$ and $B$ be non-empty sets. Let $P$ be the property on elements of $A$ 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. Now, suppose $A$ is a topological space, and $B$ is an open neighborhood of a point $x\in A$. Let $P_B$ be the property that a point of $A$ is 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 $(x_i)$). |
| \begin{enumerate} |
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| \item For example, let $A$ and $B$ be non-empty sets. Let $P$ be the property on elements of $A$ 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. |
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| \item Now, suppose $A$ is a topological space, and $B$ is an open neighborhood of a point $x\in A$. Let $P_B$ be the property that a point of $A$ is 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 $(x_i)$). |
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| \item If $A$ is a poset and $B=\lbrace x\rbrace \subseteq A$. Let $P$ be the property that an element of $A$ is $x$. Let $(x_i)$ be a net that eventually has property $P$. In particular, let $i\in D$ be the set of positive integers. Then we have a characterization of the ascending chain condition in $A$. |
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| \end{enumerate} |
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| \textbf{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 $P$. |
\textbf{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 $P$. |
