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:

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

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

• 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	2013-03-22 12:06:51
Last modified on	2013-03-22 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