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单词 LimitPoint
释义

limit point


Let X be a topological spaceMathworldPlanetmath, and let AX. An element xX is said to be a limit pointMathworldPlanetmathPlanetmath 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{x} converging to x.

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更新时间:2025/5/5 2:26:16