nLab
paracompact topological space

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

Contents

Definition

Recall:

Definition

(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 each 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. ) by a locally finite open cover (def. ).

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. ).

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.

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 Čech 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, 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

Last revised on May 17, 2020 at 02:52:39. See the history of this page for a list of all contributions to it.