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
Canonical name	TypesOfLimitPoints
Date of creation	2013-03-22 14:37:50
Last modified on	2013-03-22 14:37:50
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	7
Author	mathcam (2727)
Entry type	Definition
Classification	msc 54A99
Defines	$\omega$ -accumulation points
Defines	condensation points
Defines	cluster points