nLab paracompact topological space

Definition

Recall:

Definition

(locally finite cover)

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

An open cover $\left\{U_i \subset X \right\}_{i \in I}$ of $X$ is called locally finite if for each point $x \in X$, there exists a neighbourhood $U_x \supset \left\{x\right\}$ such that it intersects only finitely many elements of the cover, hence such that $U_x \cap U_i \neq \emptyset$ for only a finite number of $i \in I$.

Definition

(refinement of open covers)

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

Then a refinement of this open cover is a set of open subsets $\{V_j \subset X\}_{j \in J}$ which is still an open cover in itself and such that for each $j \in J$ there exists an $i \in I$ with $V_j \subset U_i$.

Now:

Definition

(paracompact topological space)

A topological space $(X,\tau)$ is called paracompact if every open cover of $X$ 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.

Example

Every locally connected locally compact topological group is paracompact (this prop.).

Proposition

locally compact and second-countable Hausdorff space are paracompact.

Proposition

locally compact and sigma-compact spaces are paracompact

Example

(Euclidean space is paracompact)

For $n \in \mathbb{N}$, then the Euclidean space $\mathbb{R}^n$, 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 $X = \coprod X_\lambda$ whose components are paracompact. As the union is disjoint, the components, that is to say, the $X_\lambda$, are open in $X$. Thus any open cover, say $\mathcal{U}$, of $X$ has a refinement by open sets, say $\mathcal{V}$, such that each $V \in \mathcal{V}$ is contained in some $X_\lambda$. Thus we can write $\mathcal{V} = \coprod \mathcal{V}_\lambda$. As each $X_\lambda$ is paracompact, each $\mathcal{V}_\lambda$ has a locally finite refinement, say $\mathcal{W}_\lambda$. Then let $\mathcal{W} := \coprod \mathcal{W}_\lambda$. As each $\mathcal{W}_\lambda$ is a refinement of the corresponding $\mathcal{V}_\lambda$, $\mathcal{W}$ is a refinement of $\mathcal{V}$, and hence of $\mathcal{U}$. As each point of $X$ has a neighbourhood which meets only elements of one of the $\mathcal{W}_\lambda$, and as that $\mathcal{W}_\lambda$ is locally finite, $\mathcal{W}$ is locally finite. Thus $\mathcal{U}$ 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 $E$ is the sequential limit of separable Hilbert spaces $H_n$, such that the canonical projections

      $$p_n : E \to H_n$$

      satisfy

      $$closure(p_n^{-1}(B)) = p_n^{-1}(closure(B))$$

      for any open ball $B$ in $H_n$, then $E$ is paracompact, and furthermore admits smooth partitions of unity.

      ( Brylinski, section I.4)

• 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

  • every separable metric space is paracompact;

  • every metric space whatsoever is paracompact, assuming the axiom of choice; see at metric spaces are paracompact

  • pseudometric spaces are paracompact under the same conditions, if one does not require Hausdorffness;

• In particular we have the following implications

  • second-countable space and regular Hausdorff space

    $\Rightarrow$ metrizable space $\Rightarrow$ paracompact space

    (the first is Urysohn’s metrization theorem, the second is due to Stone 48, see also at second-countable regular spaces are paracompact and metric spaces are paracompact)

  • paracompact space and locally metric space $\Rightarrow$ metrizable space

    (this is due to Smirnov)

• 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

Proposition

A closed subspace of a paracompact topological space is itself paracompact.

(e.g. here);

• Dieudonne’s theorem: paracompact Hausdorff spaces are normal

• every paracompact finite-dimensional topological manifold has a partition of unity

• paracompact Hausdorff spaces equivalently admit subordinate partitions of unity

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

• For paracompact Hausdorff spaces, all open covers are numerable open covers.

• Michael's theorems

Lemma

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

$$\{V_n \subset X\}_{n \in \mathbb{N}}$$

such that each element $V_n$ is a union of open subsets of $X$ each of which is contained in at least one of the elements $U_i$ of the original cover.

(e.g. Hatcher, lemma 1.21)

Proof

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

For $J \subset I$ a finite set, let

$$V_J \;\coloneqq\; \left\{ x \in X \;\vert\; \underset{j \in J}{\forall} \left( \underset{k \in I \setminus J}{\forall} \left( f_j(x) \gt f_k(x) \right) \right) \right\} \,.$$

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

Moreover, $V_J$ is contained in $supp(f_j)$ for $j \in J$ and hence in one of the $U_i$.

Now for $n \in \mathbb{N}$ take

$$V_n \;\coloneqq\; \underset{ {J \subset I} \atop { {\vert J\vert} = n } }{\cup} V_J$$

to be the union of the $V_J$ over all subset $J$ with precisely $n$ elements.

The set $\{V_n \subset X\}_{n \in \mathbb{N}}$ is a cover because for any $x \in X$ we have $x \in V_{J_x}$ for

$$J_x \coloneqq \{ i \in I \;\vert\; f_i(x) \gt 0 \}$$

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

Colimits

See at colimits of paracompact Hausdorff spaces.

Homotopy and Cohomology

• On paracompact spaces, abelian Čech cohomology does compute abelian sheaf cohomology,

  i.e. the canonical morphism $\check{H}(X,A) \to H(X,A)$ for $A$ any chain complex of sheaves is an isomorphism when the topological space underlying $X$ is paracompact.

Proposition

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

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

Lemma

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

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

This appears as (HTT, lemma 7.2.3.5).

Related concepts

• countably paracompact topological space

• compact topological space, countably compact topological space, locally compact topological space, strongly compact topological space, sequentially compact topological space

• orthocompact space

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

• R. Engelking, General topology, chapter 5 is dedicated to paracompact spaces

• Brian Conrad: Paracompactness and local compactness [pdf, pdf]

• D. K. Burke, Covering properties, in: K. Kunen, J.E. Vaughan (eds.), Handbook of Set-Theoretic Topology, North-Holland (1984) Ch. 9, 347–422

• Allen Hatcher, section 1.2 of Vector bundles & K-theory (web)

• Heikki Junnila, pp. 73 in: A second course in general topology (2007) [pdf]

• English Wikipedia: paracompact space

• Springer eom: paracompact space, paracompactness criteria

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

• Jean-Luc Brylinski, Loop spaces, characteristic classes geoemetric quantization

• 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