nLab
Michael's theorem

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

There are several theorems of Ernest Michael from the 1950s about paracompactness. There is also Michael’s 1972 characterization of paracompact locally compact spaces under certain class of quotient maps.

Statement

Proposition

(detection of paracompactness, Michael 53, theorem 1)

Let XX be a topological space such that

  1. XX is regular;

  2. every open cover of XX has a refinement by a union of a countable set of

    locally finite sets of open subsets (not necessarily covering).

Then XX is paracompact topological space.

Prop. 1 immediately implies that

Proposition

(on the closed image of a paracompact space, Michael 57, corollary 1)

The image of a paracompact Hausdorff space under a closed continuous function is also paracompact Hausdorff.

Proposition

(Michael selection theorem)

A lower semicontinuous map from a paracompact topological space XX to a Banach space EE with convex closed values has a continuous subrelation which is a function. If this is true for a given topological space YY instead of EE and all such functions and codomains EE, then YY is paracompact.

References

The original articles are the following:

See also:

Revised on May 29, 2017 16:00:07 by Urs Schreiber (178.6.236.87)