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eventual property (Definition)

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$, $ 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
    $\displaystyle S=\lbrace k \in D \mid (x_i)$ eventually has $\displaystyle P_k\rbrace.$
    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$.



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Other names:  residually constant
Also defines:  eventually, directed net, eventually constant
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Cross-references: class, fixed, image, preorder, ascending chain condition, chain, poset, accumulation point, characterization, point, neighborhood, open, topological space, eventually coincide, intersection, directed set, net, property
There are 49 references to this entry.

This is version 13 of eventual property, born on 2007-01-15, modified 2007-06-12.
Object id is 8770, canonical name is EventualProperty.
Accessed 2354 times total.

Classification:
AMS MSC: 06A06 (Order, lattices, ordered algebraic structures :: Ordered sets :: Partial order, general)
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