nLab numerable open cover

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 $\{U_i\}$ be an open cover of the topological space $X$ (actually Dold 1963 doesn’t always require open, see discussion below). It is said to be numerable if there is a collection of continuous functions $\phi_i \colon X \to [0,1]$ (to the closed interval) such that:

1. the support of the $i$th function is contained in the $i$th subset:

   $Supp(\phi_i) \subset U_i$;

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

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

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

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

Alternative characterizations

Proposition

(Ernest Michael, Kiiti Morita, Arthur H. Stone, see Morita 1962, Thm. 1.2)

For an open cover $\{U_i\}_{i\in I}$ of a topological space $X$ the following properties are equivalent.

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

• (Michael 1953, Prop. 2) $U$ admits a subordinate locally finite partition of unity;

• (Stone, Thms. 1, 2.) $U$ is a normal cover, meaning there is a sequence $W_0=U$, $W_1$, $W_2$, $\ldots$ of open covers of $X$ such that $W_{n+1}$ is a star refinement? of $W_n$ for all $n\ge0$;

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

• (Mardešić & Segal 1982, Lemma I.6.1) $U$ can be refined by the inverse image of an open cover of $Y$ under some continuous map $X\to Y$, where $Y$ is an absolute neighborhood retract, i.e., a metrizable topological space $Y$ such that any closed embedding $Y\to Z$ into a metrizable topological space $Z$ factors through an open subset $U\subset Z$ such that there is a map $U\to Y$ for which the composition $Y\to U\to Y$ is identity;

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

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

• (Michael 1953, Thm. 1, Morita 1964, Thm. 1.2.) $U$ can be refined by an open cover given by the union of a countable collection of locally finite families of cozero sets.

• (Michael 1953, Proposition 1, Hoshina 1989, Theorem 1.1 and 1.2) $U$ 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 $\mathcal{B}G$ 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:10.1090/S0002-9939-1991-1054165-6$]$

• Albrecht Dold, Partitions of unity in the theory of fibrations, Ann. of Math. 78 2 (1963) 223–255 $[$MR0155330 jstor:1970341 doi:10.2307/1970341$]$

• Kiiti Morita, Products of normal spaces with metric spaces, Mathematische Annalen 154:4 (1964), 365–382 $[$doi:10.1007/BF01362570$]$

• 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

• Albrecht Dold, Lectures on algebraic topology

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.

Detailed review of numerable covers and discussion of descent over them for simplicial presheaves via a Brown-Gersten property:

• Dmitri Pavlov, Numerable open covers and representability of topological stacks, Topology and its Applications 318 (2022) 108203 [arXiv:2203.03120, doi:10.1016/j.topol.2022.108203]