# 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. 1.

unilateral condensation point: $x$ is a condensation point of $A$ and there is a positive $\epsilon$ with either $(x-\epsilon,x)\cap A$ countable or $(x,x+\epsilon)\cap A$ countable.

2. 2.

bilateral condensation point: For all $\epsilon>0$, we have both $(x-\epsilon,x)\cap A$ and $(x,x+\epsilon)\cap A$ 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 CondensationPoint 2013-03-22 16:40:45 2013-03-22 16:40:45 sauravbhaumik (15615) sauravbhaumik (15615) 9 sauravbhaumik (15615) Definition msc 54A05