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4
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'net'
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| Title of object: |
net |
| Canonical Name: |
Net |
| Type: |
Definition |
| Created on: |
2002-08-01 11:58:03 |
| Modified on: |
2005-07-15 14:19:13 |
| Classification: |
msc:54A20 |
| Defines: |
subnet |
Revision comment (for changes between this and next version):
| Changes for correction #8173 ('missing $$'). |
Preamble:
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% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
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\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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%\usepackage{amsthm}
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%\usepackage{xypic}
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Content:
Let $X$ be a set. A \emph{net} is a map from a directed set to X. In other words, it is a pair $(A,\gamma)$ where $A$ is a directed set and $\gamma$ is a map from $A$ to $X$. If $a\in A$ then $\gamma(a)$ is normally written $x_a$, and then the net is written $(x_a)_{a\in A}$.
Now suppose $X$ is a topological space, $A$ is a directed set, and $(x_a)_{a\in A}$ is a net. Let $x\in X$. $(x_a)$ is said to \emph{converge} to $x$ iff whenever ever $U$ is an open neighbourhood of $x$, there is some $b \in A$ such that $x_a \in U$ whenever $a \geq b$..
Similarly, $x$ is said to be an \emph{accumulation point} of $(x_a)$ iff whenever $U$ is an open neighbourhood of $x$ and $b \in A$ there is $a \in A$ such that $a \geq b$ and $x_a \in U$.
Now let $B$ be another directed set, and let $\delta :B\rightarrow A$ be an increasing map such that $\delta(B)$ is cofinal in $A$. Then the pair $(B, \gamma\circ\delta)$ is said to be a \emph{subnet} of $(A,\gamma)$.
Under these definitions, nets become a generalisation of sequences to arbitrary topological spaces. For example:
\begin{itemize}
\item
if $X$ is Hausdorff then any net in $X$ converges to at most one point
\item
if $Y$ is a subspace of $X$ then $x\in\overline{Y}$ iff there is a net in $Y$ converging to $x$
\item
if $X'$ is another topological space and $f:X\rightarrow X'$ is a map then $f$ is continuous at $x$ iff whenever $(x_a)$ is a net converging to $x$, $(f(x_a))$ is a net converging to $f(x)$
\item
$X$ is compact iff every net has a convergent subnet
\end{itemize} |
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