nLab
paracompact topological space

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

Contents

Definition

Recall:

Definitione

(locally finite cover)

Let (X,τ)(X,\tau) be a topological space.

An open cover {U iX} iI\{U_i \subset X\}_{i \in I} of XX is called locally finite if for all point xXx \in X, there exists a neighbourhood U x{x}U_x \supset \{x\} such that it intersects only finitely many elements of the cover, hence such that U xU iU_x \cap U_i \neq \emptyset for only a finite number of iIi \in I.

Definition

(refinement of open covers)

Let (X,τ)(X,\tau) be a topological space, and let {U iX} iI\{U_i \subset X\}_{i \in I} be a open cover.

Then a refinement of this open cover is a set of open subsets {V jX} jJ\{V_j \subset X\}_{j \in J} which is still an open cover in itself and such that for each jJj \in J there exists an iIi \in I with V jU iV_j \subset U_i.

Now:

Definition

(paracompact topological space)

A topological space (X,τ)(X,\tau) is called paracompact if every open cover of XX has a refinement (def. 2) by a locally finite open cover (def. 1).

Remark

(differing terminology)

As with the concept of compact topological spaces (this remark), some authors demand a paracompact space to also be a Hausdorff topological space. See at paracompact Hausdorff space.

Examples

Example

Every compact space is paracompact.

Proposition

locally compact and second-countable Hausdorff space are paracompact.

Example

(Euclidean space is paracompact)

For nn \in \mathbb{N}, then the Euclidean space n\mathbb{R}^n, regarded with its metric topology is paracompact.

Proof

Euclidean space is evidently locally compact and sigma-compact. Therefore the statement follows since locally compact and sigma-compact spaces are paracompact (prop. 2).

Proposition

Paracompactness is preserved by forming disjoint union spaces (coproducts in Top).

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 smooth loop space 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.

  • CW-complexes are paracompact Hausdorff spaces (Miyazaki 52), see for instance Hatcher, appendix of section 1.2. For a discussion that highlights which choice principles are involved, see (Fritsch-Piccinini 90, Theorem 1.3.5 (p. 29 and following)).

  • 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 connected) or metrisable.

    • the Sorgenfrey plane (a product of two Sorgenfrey lines) is not paracompact. This shows that the product of paracompact spaces need not be paracompact.

Properties

General

Lemma

Let XX be a paracompact Hausdorff space, and let {U iX} iI\{U_i \subset X\}_{i \in I} be an open cover. Then there exists a countable cover

{V nX} n \{V_n \subset X\}_{n \in \mathbb{N}}

such that each element V nV_n is a union of open subsets of XX each of which is contained in at least one of the elements U iU_i of the original cover.

(e.g. Hatcher, lemma 1.21)

Proof

Let {f i:X[0,1]} iI\{f_i \colon X \to [0,1]\}_{i \in I} be a partition of unity subordinate to the original cover, which exists since paracompact Hausdorff spaces equivalently admit subordinate partitions of unity.

For JIJ \subset I a finite set, let

V J{xX|jJ(kIJ(f j(x)>f k(x)))}. V_J \;\coloneqq\; \left\{ x \in X \;\vert\; \underset{j \in J}{\forall} \left( \underset{k \in I \setminus J}{\forall} \left( f_j(x) \gt f_k(x) \right) \right) \right\} \,.

By local finiteness there are only a finite number of f k(x)f_k(x) greater than zero, hence the condition on the right is a finite number of strict inequalities. Since the f if_i are continuous, this implies that V JV_J is an open subset.

Moreover, V JV_J is contained in supp(f j)supp(f_j) for jJj \in J and hence in one of the U iU_i.

Now for nn \in \mathbb{N} take

V nJI|J|=nV J V_n \;\coloneqq\; \underset{ {J \subset I} \atop { {\vert J\vert} = n } }{\cup} V_J

to be the union of the V JV_J over all subset JJ with precisely nn elements.

The set {V nX} n\{V_n \subset X\}_{n \in \mathbb{N}} is a cover because for any xXx \in X we have xV J xx \in V_{J_x} for

J x{iI|f i(x)>0} J_x \coloneqq \{ i \in I \;\vert\; f_i(x) \gt 0 \}

(which is finite by local finitness of the partition of unity).

Colimits

See at colimits of paracompact Hausdorff spaces.

Homotopy 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

The notion of paracompact space was introduced in

  • Jean Dieudonné, Une généralisation des espaces compacts, Journal de Mathématiques Pures et Appliquées, Neuvième Série, 23: 65–76 (1944)

That fully normal spaces are equivalently paracompact is due to

  • A. H. Stone, Paracompactness and product spaces, Bull. Amer. Math. Soc. Volume 54, Number 10 (1948), 977-982. (Euclid)

General accounts include

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

  • Rudolf Fritsch and Renzo A. Piccinini, Cellular structures in topology, Cambridge studies in advanced mathematics Vol. 19, Cambridge University Press (1990). (pdf)

Discussion of paracompactness of CW-complexes includes

  • Hiroshi Miyazaki, The paracompactness of CW-complexes, Tohoku Math. J. (2) Volume 4, Number 3 (1952), 309-313. 1952 Euclid

Revised on June 6, 2017 04:48:43 by Urs Schreiber (88.77.226.246)