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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 $ \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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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
There are 5 references to this entry.

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 5341 times total.

Classification:
AMS MSC: 54A99 (General topology :: Generalities :: Miscellaneous)
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