The space of ANR’s of a closed surface

Laurence Richard Boxer

Vol. 79 (1978), No. 1, 47–68

DOI: 10.2140/pjm.1978.79.47

Abstract

We study the hyperspace (denoted 2_h^M) of ANR’s of a (polyhedral) closed surface M. The topology of 2_h^M is induced by Borsuk’s homotopy metric. We show the subpolyhedra of M are dense in 2_h^M. We obtain a necessary and sufficient condition for an arc in 2_h^M joining two points. We show that 2_h^M is an ANR (ℳ). We prove that the subspace of 2_h^M whose members are AR’s has the homotopy type of M.

Mathematical Subject Classification 2000

Primary: 54C55

Secondary: 57N99

Milestones

Received: 26 April 1977

Revised: 5 January 1978

Published: 1 November 1978