Michael's theorem

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

Theorem. (Michael theorem on the closed image of a paracompact) An image of a paracompact Hausdorff space under a closed continuous map is also paracompact Hausdorff.

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.

  • Ernest Michael, Proc. Amer. Math. Soc. 1953

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

  • E, Michael, Another note on paracompactness, 1957, pdf

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

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

  • R. Engelking, General topology

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

  • wikipedia Michael selection theorem

Created on June 22, 2011 13:13:07 by Zoran Škoda (