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

2. bilateral condensation point: For all $ϵ>0$, we have both $(x-ϵ,x)\cap A$ and $(x,x+ϵ)\cap A$ 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.