paracompact topological space



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.


  • locally compact spaces

  • paracompactness is preserved by disjoint union (coproduct).


    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


        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)


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



        2. Each component of MM is second countable.
        3. MM is metrisable.
        4. MM is paracompact.


        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.







  • 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.



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


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.


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


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 (