paracompact topological space




topology (point-set topology, point-free topology)



Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory





(locally finite cover)


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.


(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.



(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. ).


(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.



Every compact space is paracompact.


locally compact and second-countable Hausdorff space are paracompact.


(Euclidean space is paracompact)

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


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


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


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.




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)


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).


See at colimits of paracompact Hausdorff spaces.

Homotopy and Cohomology


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.


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


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 March 22, 2021 at 10:56:41. See the history of this page for a list of all contributions to it.