nLab
paracompact topological space

Contents

Definition

A topological space XX is called paracompact if every open cover {U iX}\{U_i \to X\} has a refinement by an open covering {V jX}\{V_j \to X\} that is locally finite, i.e. such that every point has a neighbourhood that intersects only finitely many of the open subsets V jV_j.

Terminological variants

The notion of paracompact space was introduced by Dieudonné in 1944. Often one requires a paracompact space to be Hausdorff as well. The term paracompactum accordingly often denotes a paracompact and Hausdorff space. This convention is in line with that for compact spaces.

Examples

  • locally compact spaces

  • paracompactness is preserved by disjoint union (coproduct).

    Proof

    Consider a disjoint union X=X λX = \coprod X_\lambda whose components are paracompact. As the union is disjoint, the components, that is to say, the X λX_\lambda, are open in XX. Thus any open cover, say 𝒰\mathcal{U}, of XX has a refinement by open sets, say 𝒱\mathcal{V}, such that each V𝒱V \in \mathcal{V} is contained in some X λX_\lambda. Thus we can write 𝒱=𝒱 λ\mathcal{V} = \coprod \mathcal{V}_\lambda. As each X λX_\lambda is paracompact, each 𝒱 λ\mathcal{V}_\lambda has a locally finite refinement, say 𝒲 λ\mathcal{W}_\lambda. Then let 𝒲:=𝒲 λ\mathcal{W} := \coprod \mathcal{W}_\lambda. As each 𝒲 λ\mathcal{W}_\lambda is a refinement of the corresponding 𝒱 λ\mathcal{V}_\lambda, 𝒲\mathcal{W} is a refinement of 𝒱\mathcal{V}, and hence of 𝒰\mathcal{U}. As each point of XX has a neighbourhood which meets only elements of one of the 𝒲 λ\mathcal{W}_\lambda, and as that 𝒲 λ\mathcal{W}_\lambda is locally finite, 𝒲\mathcal{W} is locally finite. Thus 𝒰\mathcal{U} has a locally finite refinement.

  • manifolds

    • finite-dimensional manifolds are locally compact, so we have the results above, but we also have some converses:

      • a finite-dimensional Hausdorff topological manifold is paracompact precisely if it is metrizable

      • a finite-dimensional Hausdorff topological manifold is paracompact precisely if each component is second-countable

    • infinite-dimensional manifolds are generally not locally compact, but we still have some results:

      • The Frechet manifold of smooth loops of a compact finite dimensional manifold is paracompact.

      • More generally, if EE is the sequential limit of separable Hilbert spaces H nH_n, such that the canonical projections

        p n:EH n p_n : E \to H_n

        satisfy

        closure(p n 1(B))=p n 1(closure(B)) closure(p_n^{-1}(B)) = p_n^{-1}(closure(B))

        for any open ball BB in H nH_n, then EE is paracompact, and furthermore admits smooth partitions of unity.

        ( Brylinski, section I.4)

        Urs Schreiber: don’t we need some extra assumption here? Otherwise why wouldn’t this imply that every space modeled on n\mathbb{R}^n is paracompact, while it is only the second-countable such that are?

        Toby: Probably Brylinski has a requirement of metrisability or something.

        Andrew Stacey: More generally, the coproduct of paracompact spaces is again paracompact.

        Spivak, in A Comprehensive Introduction to Differential Geometry, I has an appendix on non-metrisable manifolds. He starts (in the appendix) by defining a manifold to be a Hausdorff locally Euclidean space. Then he proves that for any manifold MM TFAE:

        1. Each component of MM is σ\sigma-compact.
        2. Each component of MM is second countable.
        3. MM is metrisable.
        4. MM is paracompact.

        The bit to highlight is the words “Each component of …” in the first two.

        Toby: Right, and I've got the ‘each component’ clause in the section on finite-dimensional manifolds. But how does this work for infinite-dimensional manifolds? (I'm also unsure how things work for non-Hausdorff manifolds, or more generally for non-Hausdorff locally compact spaces.

        David Roberts: Lang has the result that second countable manifolds are paracompact. For him manifold seems to mean that the space is Hausdorff and locally euclidean, with no restriction on the type of vector space. Further, smooth partitions of unity exist. This is Corollary 3.4 in the 2002 edition of Introduction to differentiable manifolds

        Andrew Stacey Surely “locally Euclidean” implies that the model space is n\mathbb{R}^n. Is this the book where he deals with finite and infinite dimensional manifolds all in one go? In that case, I would caution that if I remember right (don’t have the book in front of me) he’s using manifolds modelled on at most Banach spaces. For smooth partitions of unity to exist then you need to know that there is a smooth bump function, which hinges on some properties of the norm. Kriegl and Michor (who else?) address this in chapter III of their book. In particular, they quote a result due to Kurzweil (1954) that C([0,1])C([0,1]) and 1\ell^1 are not C 1C^1-regular.

        To Toby, I guess that the issue about components for infinite dimensional manifolds is dealt with in what I did before. The question reduces to figuring out if a specific component is paracompact. To use “metrisable implies paracompact” you’ve got to be on a Frechet manifold (since Frechet is the limit of metrisability). Brylinski’s construction outlined about is sufficient to ensure that (countable family of semi-norms). To have smooth partitions of unity, you then need smoothly regular. If the semi-norms are smooth (away from zero) then, obviously, that’s sufficient. In Brylinski’s construction then that comes from the fact that the semi-norms are defined by Hilbertian norms. I’m not sure what the condition on the closures is for, can anyone scan through the proof and see why they are used? For the proof of paracompactness and partitions of unity then I don’t see immediately why they are needed. I expect I’m being dense? but if someone could quickly enlighten me, I’d be grateful.

        (Incidentally, since this query box is contained within a list, it’s important that all paragraphs are indented properly, otherwise strange things happen)

        David Roberts: When I said locally Euclidean, I was being lazy. I meant modelled on some vector space. And yes Lang does finite and infinite dimensions at the same time. Regarding the condition Brylinski uses, he says it implies that the image of p n:EH np_n : E \to H_n from the ILH space EE to each of the Hilbert spaces H nH_n in the sequence is dense. I don’t know how this is used, I’m relying on G00gle books. Ah, but now that I check, Lang also says unless specified, vector space will mean finite dimensional vector space

        Andrew Stacey Hmm, I guess I’ll have to get Lang and Brylinski out of the library to see exactly what’s going on here. I don’t see how density can make any difference since we could always restrict to the closure of p n(E)p_n(E) in H nH_n.

  • metric spaces

  • In particular we have the following implications

  • special cases

    • the Sorgenfrey line? is a good example of a paracompact space that doesn't fit into other general classes of paracompact spaces (in particular, it is not metrisable, locally compact, or a manifold);
  • counterexamples

    • the long line is not paracompact, even though it is a manifold (unless one specifically requires paracompactness of manifolds) but it fails to be second-countable (even though it is connnected) or metrisable.

Properties

General

Care should be taken as to which category one constructs partitions of unity on paracompact spaces. For example, analytic partitions of unity generally do not exist on smooth (finite dimensional) manifolds, even when smooth ones do.

Homomotopy and Cohomology

Proposition

On a paracompact space XX, every hypercover of finite height is refined by the Cech nerve of an ordinary open cover.

For a hypercover of height nn \in \mathbb{N}, this follows by intersecting the open covers that are produced by the following lemma for 0kn0 \leq k \leq n.

Lemma

For XX a paracompact topological space, let {U α} αA\{U_\alpha\}_{\alpha \in A} be an open cover, and let each (k+1)(k+1)-fold intersection U α 0,,α kU_{\alpha_0, \cdots, \alpha_{k}} be equipped itself with an open cover {V β α 0,,α k}\{V^{\alpha_0, \cdots, \alpha_k}_{\beta}\}.

Then there exists a refinement {U α}\{U'_{\alpha'}\} of the original cover, such that each (k+1)(k+1)-fold intersection U α 0,,α kU'_{\alpha'_0, \cdots, \alpha'_k} for all indices distinct is contained in one of the V βV_\beta.

This appears as (HTT, lemma 7.2.3.5).

References

A basic discussion with an eye towards abelian sheaf cohomology and abelian Čech cohomology is around page 32 of

Revised on May 17, 2012 19:24:10 by Toby Bartels (64.89.53.19)