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limit point
Let be a topological space, and let . An element is said to be a limit point of if every open set containing also contains at least one point of distinct from . Note that we can often take a nested sequence of open such sets, and can thereby construct a sequence of points which converge to , partially motivating the terminology ”limit” in this case.
Equivalently:
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is a limit point of if and only if there is a net in converging to which is not residually constant.
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is a limit point of if and only if there is a filter on converging to .
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If is a metric (or first countable) space, is a limit point of if and only if there is a sequence of points in converging to .
Keywords:
topology
Related:
AlternateStatementOfBolzanoWeierstrassTheorem
Synonym:
accumulation point, cluster point
Type of Math Object:
Definition
Major Section:
Reference
Mathematics Subject Classification
54A99 None of the above, but in MSC2010 section 54Axx- Forums
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