condensation point
Let $X$ be a topological space^{} and $A\subset X$. A point $x\in X$ is called a condensation point of $A$ if every open neighbourhood of $x$ contains uncountably many points of $A$.
For example, if $X=\mathbb{R}$ and $A$ any subset, then any accumulation point^{} of $A$ is automatically a condensation point. But if $X=\mathbb{Q}$ and $A$ any subset, then $A$ does not have any condensation points at all.
We have further classifications of condensation point where the topological space is an ordered field. Namely,
 1.

2.
bilateral condensation point: For all $\u03f5>0$, we have both $(x\u03f5,x)\cap A$ and $(x,x+\u03f5)\cap A$ uncountable.
If $\kappa $ is any cardinal (i.e. an ordinal^{} which is the least among all ordinals of the same cardinality as itself), then a $\kappa $condensation point can be defined similarly.
Title  condensation point 

Canonical name  CondensationPoint 
Date of creation  20130322 16:40:45 
Last modified on  20130322 16:40:45 
Owner  sauravbhaumik (15615) 
Last modified by  sauravbhaumik (15615) 
Numerical id  9 
Author  sauravbhaumik (15615) 
Entry type  Definition 
Classification  msc 54A05 