Edge of the World: When Are Manifolds Metrisable?

Chapter

Abstract

This chapter might seem odd in that it lists a huge number of topological properties and connections between them. What it shows is that the requirement that a manifold be metrisable is extremely versatile. We list over 100 conditions each of which is equivalent to metrisability of a manifold. At one extreme, metrisability of a manifold implies that it may be embedded as a closed subset of some Euclidean space while at the other extreme knowing that every open cover of the form \(\{U_{\alpha }\ /\ {\alpha }<{\omega }_1\}\) with \(U_{\alpha }\subset U_{\beta }\) whenever \({\alpha }<{\beta }\) has an open refinement which is point countable on a dense subset is sufficient to guarantee that a manifold is metrisable. Space precludes giving full details of the proofs. Instead we give brief ideas of the proofs and refer the interested reader to original sources for complete proofs. The content of this chapter is taken from [21].

Keywords

Topological Space Closed Subset Open Cover Dense Subset Countable Collection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Singapore 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AucklandAucklandNew Zealand

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