# 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.

Title types of limit points TypesOfLimitPoints 2013-03-22 14:37:50 2013-03-22 14:37:50 mathcam (2727) mathcam (2727) 7 mathcam (2727) Definition msc 54A99 $\omega$-accumulation points condensation points cluster points