There is a continuum which is connected by uniformly short paths but not uniformly path connected

Melvin R. Holmes

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

David Bellamy defined the concept of connectedness by uniformly short paths and asked wether it was equivalent to the notion of uniform path connectedness. A counterexample is described. Some definitions: for a positive real number e, a Peano continuum has "e-length" ≤ an integer n iff it is the union of a chain of n Peano continua of diameter less than e. A continuum (compact connected metric space) X is "uniformly path connected" iff there is a family P of paths such that for each a, b in X, there is a path in P from a to b, and for each e there is an N such that the e-length of the image of each path in P is ≤ N. A continuum is "connected by uniformly short paths" iff for each e there is a number N such that for each pair of points a, b in X there is a path from a to b whose image has e-length ≤ N. (The definition of e-length given is more concise than that given in the paper).

Original languageEnglish
Pages (from-to)17-23
Number of pages7
JournalTopology and its Applications
Volume42
Issue number1
DOIs
StatePublished - 12 Nov 1991

Keywords

  • Connected
  • connected by uniformly short paths
  • continuum
  • path connected
  • uniformly path connected

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