# Homeomorphism Groups and Metrisation of Manifolds

### New Zealand Journal of Mathematics

Vol. 42, (2012), Pages 37-43

David Gauld

Department of Mathematics

University of Auckland,

Auckland,

New Zealand

Jan van Mill

Faculty of Sciences

Department of Mathematics

VU University Amsterdam

1081 HV Amsterdam

The Netherlands

Abstract We prove that a manifold M is metrisable if and only if its group of homeomorphisms $\mathcal{H}(M)$ endowed with the compact-open topology is a q-space. We also discuss pseudo-character and tightness.

Keywords homeomorphism group, manifold, metrisation.

Classification (MSC2000) 54C35, 54E35.