nLab numerable open cover




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

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


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory

Topos Theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory




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


Let {U i}\{U_i\} be an open cover of the topological space XX (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 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


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

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 1953, Prop. 2) UU admits a subordinate locally finite partition of unity;

  • (Stone, Thms. 1, 2.) UU is a normal cover, meaning there is a sequence W 0=UW_0=U, W 1W_1, W 2W_2, \ldots 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 & Segal 1982, Lemma I.6.1) 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 1953, Thm. 1, Morita 1964, Thm. 1.2.) UU 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) UU can be refined by an open cover given by the union of a countable collection of discrete families of cozero sets.


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.


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.

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

Last revised on August 7, 2022 at 21:43:42. See the history of this page for a list of all contributions to it.