nLab
numerable open cover

Contents

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

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

A numerable open cover (alias normal 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 continuous functions ϕ i:X[0,1]\phi_i \colon X \to [0,1] (to the closed interval) such that:

  1. the support of the iith function is contained in the iith subset:

    Supp(ϕ i)U iSupp(\phi_i) \subset U_i;

  2. at each point xXx\in X, only a finite number of the ϕ i\phi_i are non-zero;

  3. xX iϕ i(x)1\underset{x \in X}{\forall} \sum_i \phi_i(x) \equiv 1 .

(The functions {ϕ i}\{\phi_i\} are a partition of unity.)

The collection of preimages ϕ i 1(0,1]\phi_i^{-1}(0,1] then constitute a locally finite cover that refines {U i}\{U_i\}.

Alternative characterizations

Proposition

(Ernest Michael, Kiiti Morita, Arthur H. Stone, see Morita \cite[Theorem 1.2]{Morita}.)
For an open cover {U i} iI\{U_i\}_{i\in I} of a topological space XX the following properties are equivalent.

  • UU is a numerable cover, i.e., it admits a compatible positive partition;

  • (Michael \cite[Proposition 2]{Michael}.) UU admits a subordinate locally finite partition of unity;

  • (Stone \cite[Theorems 1 and 2]{Stone}.) UU is a normal cover, meaning there is a sequence W 0=UW_0=U, W 1W_1, W 2W_2, … of open covers of XX such that W n+1W_{n+1} is a star refinement? of W nW_n for all n0n\ge0;

  • UU can be refined by the inverse image of an open cover of YY under some continuous map XYX\to Y, where YY is a metrizable topological space;

  • (Mardešić and Segal \cite[Lemma I.6.1]{MardesicSegal}.) UU can be refined by the inverse image of an open cover of YY under some continuous map XYX\to Y, where YY is an absolute neighborhood retract, i.e., a metrizable topological space YY such that any closed embedding YZY\to Z into a metrizable topological space ZZ factors through an open subset UZU\subset Z such that there is a map UYU\to Y for which the composition YUYY\to U\to Y is identity;

  • UU can be refined by a locally finite normal open cover;

  • UU can be refined by a locally finite open cover consisting of cozero sets;

  • (Michael \cite[Theorem 1]{Michael}, Morita \cite[Theorem 1.2]{Morita2}.) UU can be refined by an open cover given by the union of a countable collection of locally finite families of cozero sets.

  • (Michael \cite[Proposition 1]{Michael}, Hoshina \cite[Theorem 1.1 and 1.2]{Hoshina}.) UU can be refined by an open cover given by the union of a countable collection of discrete families of cozero sets.

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 topological 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 a 1944 result by Dieudonnne that numerable covers are cofinal in locally finite covers of normal spaces — need to add this! See, eg, Theorem 6.3 of Howes’ Modern analysis and topology.

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

  • Arthur H. Stone,

    Paracompactness and product spaces, Bulletin of the American Mathematical Society 54:10 (1948), 977–983 (doi:10.1090/S0002-9904-1948-09118-2)

  • Ernest Michael,

    A note on paracompact spaces. Proceedings of the American Mathematical Society 4:5 (1953), 831–838. doi:10.1090/s0002-9939-1953-0056905-8.

  • Kiiti Morita,

    Paracompactness and product spaces. Fundamenta Mathematicae 50:3 (1962), 223–236. doi:10.4064/fm-50-3-223-236.

  • Kiiti Morita,

    Products of normal spaces with metric spaces, Mathematische Annalen 154:4 (1964), 365–382. doi:10.1007/bf01362570.

  • Albrecht Dold, Partitions of unity in the theory of fibrations, Ann. of Math. (2) 78 1963 223–255 MR0155330 jstor doi

  • Sibe Mardešić, Jack Segal,

    Shape Theory. North-Holland Mathematical Library 26 (1982).

  • Takao Hoshina, Extensions of mappings II. Topics in General Topology, Elsevier (1989).

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.

Last revised on October 5, 2021 at 02:04:08. See the history of this page for a list of all contributions to it.