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'eventual property'
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| Title of object: |
eventual property |
| Canonical Name: |
EventualProperty |
| Type: |
Definition |
| Created on: |
2007-01-15 14:57:23 |
| Modified on: |
2007-01-17 13:42:34 |
| Classification: |
msc:06A06 |
| Defines: |
eventually, directed net |
Preamble:
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Content:
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 $j\in D$.
\textbf{Examples}.
\begin{enumerate}
\item 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.
\item 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 $(x_i)$).
\item 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 \emph{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 $A$.
\item Let $R$ be a preorder and let $(x_i)_{i\in D}$ be a net in $R$. Let $A$ be the image of the net: $A=\lbrace x_i\in R \mid i\in D\rbrace$. Given a fixed $k\in D$ and some $x\in A$, let $P_k(x)$ be the property (on $A$) that $x_k\le x$. 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 \emph{directed}, or that $(x_i)$ is a \emph{directed net}. In other words, a directed net is a net $(x_i)_{i\in D}$ such that for \emph{every} $i\in D$, there is a $k(i)\in D$, such that $x_i\le x_j$ for all $j\ge k(i)$.
\end{enumerate}
\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 $P$. |
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