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Viewing Version
2
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'discrete'
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| Title of object: |
discrete |
| Canonical Name: |
Discrete2 |
| Type: |
Definition |
| Created on: |
2008-03-26 17:57:32 |
| Modified on: |
2008-03-26 18:06:40 |
| Classification: |
msc:54A05 |
Revision comment (for changes between this and next version):
| Changes for correction #13707 ('some suggestions'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
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Content:
A subset $S$ of a topological space ${\mathcal T}$ is said to be discrete if any of the following equivalent conditions is met:
\begin{itemize}
\item The subspace topology on $S$ \PMlinkescapetext{induced} by the topology on $\mathcal T$ is the discrete topology.
\item $\forall x\in S$, $\exists U\subset {\mathcal T}$ neighborhood of $x$, such that $U\cap S=\{x\}$.
\item (Only applies if ${\mathcal T}$ is a metric space) If for all sequences $(x_i)_{i\in{\mathbb N}} \in S$ that converge to some $x\in S$, there exists $N_0\in\mathbb N$ such that $\forall i\ge N_0$, $x_i=x$.
\end{itemize} |
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