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limit point (Definition)

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 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$.



"limit point" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: alternate statement of Bolzano-Weierstrass theorem

Other names:  accumulation point, cluster point
Keywords:  topology
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Cross-references: first countable, metric, filter, residually constant, net, limit, converge, sequence, point, contains, open set, topological space
There are 39 references to this entry.

This is version 11 of limit point, born on 2002-01-04, modified 2007-12-17.
Object id is 1240, canonical name is LimitPoint.
Accessed 19530 times total.

Classification:
AMS MSC: 54A99 (General topology :: Generalities :: Miscellaneous)
Pending Errata and Addenda
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