condensation point

Let X be a topological spaceMathworldPlanetmath and AX. A point xX is called a condensation point of A if every open neighbourhood of x contains uncountably many points of A.

For example, if X= and A any subset, then any accumulation pointPlanetmathPlanetmath of A is automatically a condensation point. But if X= 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 ϵ with either (x-ϵ,x)A countableMathworldPlanetmath or (x,x+ϵ)A countable.

  2. 2.

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

    If κ is any cardinal (i.e. an ordinalMathworldPlanetmathPlanetmath which is the least among all ordinals of the same cardinality as itself), then a κ-condensation point can be defined similarly.

Title condensation point
Canonical name CondensationPoint
Date of creation 2013-03-22 16:40:45
Last modified on 2013-03-22 16:40:45
Owner sauravbhaumik (15615)
Last modified by sauravbhaumik (15615)
Numerical id 9
Author sauravbhaumik (15615)
Entry type Definition
Classification msc 54A05