Abstract topological dynamics involving set-valued functions

Chris Good, Sina Greenwood, Nazli Uresin

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Abstract

Continuous functions over compact Hausdorff spaces have been completely characterised. We consider the more general problem: given a set-valued function T from an arbitrary set X to itself, does there exist a compact Hausdorff topology on X with respect to which T is upper semicontinuous? We give conditions that are necessary for T to be upper semicontinuous and point-closed if X is a compact Hausdorff space. We show that it is always possible to provide X with a compact T1 topology with respect to which T is lower semicontinuous, and consequently, if T:X - X is a function, then it is always possible to provide X with a compact T1 topology with respect to which T is continuous.
Original languageEnglish
Article number107240
Number of pages10
JournalTopology and its Applications
Volume279
Early online date4 May 2020
DOIs
Publication statusPublished - 1 Jul 2020

Keywords

  • topological dynamical system
  • T_1
  • upper semicontinuous
  • lower semicontinuous
  • compact
  • Hausdorff
  • dynamical system

ASJC Scopus subject areas

  • Geometry and Topology

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