limit point
Let $X$ be a topological space^{}, and let $A\subseteq X$. An element $x\in X$ is said to be a limit point^{} of $A$ if every open set containing $x$ also 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:

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$x$ is a limit point of $A$ if and only if there is a net in $A$ converging to $x$ which is not residually constant.

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$x$ is a limit point of $A$ if and only if there is a filter on $A$ converging (http://planetmath.org/filter) to $x$.

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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 $x$.
Title  limit point 

Canonical name  LimitPoint 
Date of creation  20130322 12:06:51 
Last modified on  20130322 12:06:51 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  15 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 54A99 
Synonym  accumulation point^{} 
Synonym  cluster point 
Related topic  AlternateStatementOfBolzanoWeierstrassTheorem 