accumulation points and convergent subnets


Let X be a topological spaceMathworldPlanetmath and (xα)αA a net in X. A point xX is an accumulation pointPlanetmathPlanetmath of (xα) if and only if some subnet of (xα) convergesPlanetmathPlanetmath to x.


Suppose first that (xαβ)βB is a subnet of (xα) converging to x. Given an open subset U of X containing x and αA, we may select β1B such that xαβU for ββ1, as well as β2B such that αβα for ββ2. Finally, because B is directed, there exists βB such that ββ1 and ββ2; we then have αβα and xαβU, so that (xα) is frequently in U, whence x is an accumulation point of (xα). Conversely, suppose that x is an accumulation point of (xα), let N be the set of open neighborhoods of x in X, directed by reverse inclusion, and let B=A×N, directed in the natural way. For each pair (γ,U)B, select α(γ,U)B such that αγ and xα(γ,U)U; (xα(γ,U))(γ,U)B is then a subnet of (xα) that converges to x, for given UN and γA, if (γ,U)(γ,U), then α(γ,U)γγ and xα(γ,U)UU. ∎

Title accumulation points and convergent subnets
Canonical name AccumulationPointsAndConvergentSubnets
Date of creation 2013-03-22 18:37:40
Last modified on 2013-03-22 18:37:40
Owner azdbacks4234 (14155)
Last modified by azdbacks4234 (14155)
Numerical id 9
Author azdbacks4234 (14155)
Entry type Theorem
Classification msc 54A20
Related topic Net
Related topic NeighborhoodMathworldPlanetmathPlanetmath
Related topic DirectedSet
Related topic CompactnessAndConvergentSubnets