nLab
numerable open cover

Context

Topology

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

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

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Extra stuff, structure, properties

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Contents

Idea

A numerable open cover is an open cover of a topological space that admits a subordinate partition of unity.

Definition

Let {U i}\{U_i\} be an open cover of the topological space XX (actually Dold doesn’t always require open, see discussion below). It is said to be numerable if there is a collection of functions ϕ i:X[0,1]\phi_i:X \to [0,1] such that

  • supp(ϕ i)¯U i\overline{supp(\phi_i)} \subset U_i,
  • at each point xXx\in X, only finitely many of the ϕ i\phi_i are non-zero,
  • iϕ i(x)1xX\sum_i \phi_i(x) \equiv 1 \forall x\in X.

The open cover ϕ i 1(0,1]\phi_i^{-1}(0,1] is then a locally finite cover that refines {U i}\{U_i\}. The functions {ϕ i}\{\phi_i\} are a partition of unity.

Properties

As a coverage, as a site

Numerable open covers form a site called the numerable site. More precisely, numerable open covers are a coverage on the category Top of topological spaces (this is essentially given in Dold’s lectures, A.2.17 , but not using this terminology).

For paracompact spaces, numerable covers are cofinal in open covers, so that the numerable site is equivalent to the open cover site for such spaces.

There is also some result by Bourbaki that I have to look up that numerable covers are cofinal in locally finite covers of normal spaces.

Relation to numerable bundles

Many classical theorems concerning fiber bundles are stated for the numerable site. For example, the classifying space G\mathcal{B}G actually classifies bundles which trivialise over a numerable cover. (References? Dold for Milnor's classifying space, and tom Dieck I think for Segal’s) These are called numerable bundles. This is because the standard constructions of the universal bundle by Minor and Segal both are trivialisable over a numerable cover.

References

The appendix of

talks about “stacked covers”: these are useful for ‘decomposing’ numerable covers of products to a sort of parameterised version depending on a numerable cover of the first factor. This is important in looking at concordance of numerable bundles.

Revised on May 26, 2017 05:32:57 by Urs Schreiber (92.218.150.85)