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'limit point'
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| Title of object: |
limit point |
| Canonical Name: |
LimitPoint |
| Type: |
Definition |
| Created on: |
2002-01-04 19:19:35 |
| Modified on: |
2007-05-07 23:16:10 |
| Classification: |
msc:54A99 |
| Keywords: |
topology |
| Synonyms: |
limit point=accumulation point
limit point=cluster point |
Revision comment (for changes between this and next version):
| Changes for correction #13333 ('contains'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
Let $X$ be a topological space, and let $A\subseteq X$. An element $x\in X$ is said to be a \emph{limit point} of $A$ if every open set containing $x$ contains at least one point of $A$ distinct from $x$. Note that we can often take a nested sequence of open such sets, and can thereby construct a sequence of points which converge to $x$, partially motivating the terminology "limit'' in this case.
Equivalently:
\begin{itemize}
\item $x$ is a limit point of $A$ if and only if there is a net in $A$ converging to $x$ which is not residually constant.
\item $x$ is a limit point of $A$ if and only if there is a filter on $A$ \PMlinkname{converging}{filter} to $x$.
\item If $X$ is a metric (or first countable) space, $x$ is a limit point of $A$ if and only if there is a sequence of points in $A\setminus\{x\}$ converging to $x$.
\end{itemize} |
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