types of limit points

Let $X$ be a topological space and $A\\subset X$ be a subset.

A point $x\\in X$ is an $\\omega$ -accumulation point of $A$ if every open set in $X$ that contains $x$ also contains infinitely many points of $A$ .

A point $x\\in X$ is a condensation point of $A$ if every open set in $X$ that contains $x$ also contains uncountably many points of $A$ .

If $X$ is in addition a metric space, then a cluster point of a sequence $\\{x_{n}\\}$ is a point $x\\in X$ such that every $\\epsilon>0$ , there are infinitely many point $x_{n}$ such that $d(x,x_{n})<\\epsilon$ .

These are all clearly examples of limit points.

单词	TypesOfLimitPoints
释义	types of limit points Let $X$ be a topological space and $A\\subset X$ be a subset. A point $x\\in X$ is an $\\omega$ -accumulation point of $A$ if every open set in $X$ that contains $x$ also contains infinitely many points of $A$ . A point $x\\in X$ is a condensation point of $A$ if every open set in $X$ that contains $x$ also contains uncountably many points of $A$ . If $X$ is in addition a metric space, then a cluster point of a sequence $\\{x_{n}\\}$ is a point $x\\in X$ such that every $\\epsilon>0$ , there are infinitely many point $x_{n}$ such that $d(x,x_{n})<\\epsilon$ . These are all clearly examples of limit points.
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