Preliminary Results

  • Fausto Di Biase
Part of the Progress in Mathematics book series (PM, volume 147)

Abstract

Let (W, ρ) be a quasi-metric space. A set D is a space of approach to (W, ρ), for the approach function \(\tilde \rho \): D x W →[0, ∞), if
  1. 1.
    for each wW there is a sequence \({\{ {z_n}\} _n}\) in D such that
    $$\mathop {\lim }\limits_{n \to \infty } \tilde \rho \left( {{z_n},w} \right) = 0$$
     
  1. 2.
    whenever
    $$\mathop {\lim }\limits_{n \to \infty } \rho \left( {{w_n},{u_n}} \right) = 0{\text{ }}and{\text{ }}\mathop {\lim }\limits_{n \to \infty } \tilde \rho \left( {{z_n},{w_n}} \right) = 0$$
    then
    $$\mathop {\lim }\limits_{n \to \infty } \tilde \rho \left( {{z_n},{u_n}} \right) = 0$$
    ;
     
  1. 3.

    whenever

     
$$\mathop {\lim }\limits_{n \to \infty } \tilde \rho \left( {{z_n},{w_n}} \right) = \mathop {\lim }\limits_{n \to \infty } \tilde \rho \left( {{z_n},{u_n}} \right) = 0$$
then
$$\mathop {\lim }\limits_{n \to \infty } \rho \left( {{w_n},{v_n}} \right) = 0$$
, where {Zn|n is a sequence in D and {wn|n, {un|n are sequences in W.

Keywords

Radon Stein Dition Dinate 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Birkhäuser 1998

Authors and Affiliations

  • Fausto Di Biase
    • 1
    • 2
  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Dip. MatematicaUniversity Roma-Tor VergataRomeItaly

Personalised recommendations