Contents

# 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 $X$ be a topological space such that

1. $X$ is regular;

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

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

Then $X$ is paracompact topological space.

Prop. 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 $X$ to a Banach space $E$ with convex closed values has a continuous subrelation which is a function. If this is true for a given topological space $Y$ instead of $E$ and all such functions and codomains $E$, then $Y$ is paracompact.

## References

The original articles are the following: