Atlas: Uniform pathwise connectedness and Whitney levels by Iwona Krzeminska

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First International Meeting on Continuum Theory
June 29 - July 1, 2000
Facultad de Ciencias Fisico-Matematicas de la Benemerita Universidad Autonoma de Puebla
Puebla, Mexico

Organizers
Raul Escobedo, Fernando Macias, Sergio Macias, Richard Schori, Carl Seaquist

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Uniform pathwise connectedness and Whitney levels
by
Iwona Krzeminska
Opole Technical University

A metric continuum X is said to be uniformly pathwise connected if there exists a family F = { p : [a, b] --> X } of paths in X , satisfying

(1) for each pair x, y in X there is a path p in F joining x with y , and

(2) for any \epsilon > 0 there is a positive integer n such that for each p in F there are numbers t₀ = a < t₁ < t₂ < ... < t_n = b such that the diameters of the images p([ t₀, t₁ ]), ... , p([ t_n-1, t_n ]) are <= \epsilon.

It is an easy observation that continuous images of the cone over the Cantor set are uniformly pathwise connected. The converse is also true and it was proved by W. Kuperberg.

We prove that the uniform pathwise connectedness is a Whitney property. The example constructed by M. R. Holmes is used to show that the uniform pathwise connectedness is not a Whitney-reversible property. This answers the questions posed by A. Illanes and S. B. Nadler, Jr. ( HYPERSPACES: Fundamentals and Recent Advances, Marcel Dekker, Inc., New York 1999, p. 250 ).

Date received: April 14, 2000