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types of limit points (Definition)

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

A point $x\in X$ is an <</SPAN>#99#>$\omega$ 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 <</SPAN>#100#>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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Also defines:  $\omega$-accumulation points, condensation points, cluster points
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Cross-references: sequence, cluster point, metric space, addition, contains, open set, point, subset, topological space
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This is version 4 of types of limit points, born on 2004-09-24, modified 2005-10-23.
Object id is 6211, canonical name is TypesOfLimitPoints.
Accessed 7487 times total.

Classification:
AMS MSC54A99 (General topology :: Generalities :: Miscellaneous)

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