limit point
Let X be a topological space, and let A⊆X. An element x∈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∖{x} converging to x.
Title | limit point |
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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 |