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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

Approach Region Full Measure Homogeneous Type Doubling Property Pseudo Convex Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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

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