nLab Michael's theorem

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

• second-countable regular spaces are paracompact,

• locally compact and sigma-compact spaces are paracompact.

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.

Related entries

• selection theorem

References

The original articles are the following:

• Ernest Michael, A note on paracompact spaces, Proc. Amer. Math. Soc. 1953 (jstor, pdf)

• Ernest Michael, Another note on paracompactness, 1957, (pdf, pdf)

• Ernest Michael, Continuous selections. I, Annals of Math. 63 (2): 361–382 (doi:10.2307/1969615, MR0077107)

• Ernest Michael, A quintuple quotient test, Gen. Topol. Appl. 2 (1972) 91–138, link

• discussion at A. Mathew’s blog and also on an application to metric spaces here

See also:

• R. Engelking, General topology

• Kenneth Kunen, Paracompactness of box products of compact spaces, Trans. Amer. Math. Soc. 240, (1978) 307-316, jstor

• Akhil Mathew, A theorem of Michael on paracompactness

• wikipedia Michael selection theorem