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In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by Dieudonné (1944). Every compact space is paracompact.[1] Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if[2] and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff.

Every closed subspace of a paracompact space is paracompact. While compact subsets of Hausdorff spaces are always closed, this is not true for paracompact subsets. A space such that every subspace of it is a paracompact space is called hereditarily paracompact. This is equivalent to requiring that every open subspace be paracompact.

The notion of paracompact space is also studied in pointless topology, where it is more well-behaved. For example, the product of any number of paracompact locales is a paracompact locale, but the product of two paracompact spaces may not be paracompact.[3][4] Compare this to Tychonoff's theorem, which states that the product of any collection of compact topological spaces is compact. However, the product of a paracompact space and a compact space is always paracompact.

Every metric space is paracompact. A topological space is metrizable if and only if it is a paracompact and locally metrizable Hausdorff space.

Definition

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A cover of a set is a collection of subsets of whose union contains . In symbols, if is an indexed family of subsets of , then is a cover of if

A cover of a topological space is open if all its members are open sets. A refinement of a cover of a space is a new cover of the same space such that every set in the new cover is a subset of some set in the old cover. In symbols, the cover is a refinement of the cover if and only if, for every in , there exists some in such that .

An open cover of a space is locally finite if every point of the space has a neighborhood that intersects only finitely many sets in the cover. In symbols, is locally finite if and only if, for any in , there exists some neighbourhood of such that the set

is finite. A topological space is now said to be paracompact if every open cover has a locally finite open refinement.

This definition extends verbatim to locales, with the exception of locally finite: an open cover of is locally finite iff the set of opens that intersect only finitely many opens in also form a cover of . Note that an open cover on a topological space is locally finite iff its a locally finite cover of the underlying locale.

Examples

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Some examples of spaces that are not paracompact include:

Properties

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Paracompactness is weakly hereditary, i.e. every closed subspace of a paracompact space is paracompact. This can be extended to F-sigma subspaces as well.[10]

Although a product of paracompact spaces need not be paracompact, the following are true:

Both these results can be proved by the tube lemma which is used in the proof that a product of finitely many compact spaces is compact.

Paracompact Hausdorff spaces

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Paracompact spaces are sometimes required to also be Hausdorff to extend their properties.

  • (Theorem of Jean Dieudonné) Every paracompact Hausdorff space is normal.
  • Every paracompact Hausdorff space is a shrinking space, that is, every open cover of a paracompact Hausdorff space has a shrinking: another open cover indexed by the same set such that the closure of every set in the new cover lies inside the corresponding set in the old cover.
  • On paracompact Hausdorff spaces, sheaf cohomology and Čech cohomology are equal.[11]

Partitions of unity

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The most important feature of paracompact Hausdorff spaces is that they admit partitions of unity subordinate to any open cover. This means the following: if X is a paracompact Hausdorff space with a given open cover, then there exists a collection of continuous functions on X with values in the unit interval [0, 1] such that:

  • for every function f: X → R from the collection, there is an open set U from the cover such that the support of f is contained in U;
  • for every point x in X, there is a neighborhood V of x such that all but finitely many of the functions in the collection are identically 0 in V and the sum of the nonzero functions is identically 1 in V.

In fact, a T1 space is Hausdorff and paracompact if and only if it admits partitions of unity subordinate to any open cover (see below). This property is sometimes used to define paracompact spaces (at least in the Hausdorff case).

Partitions of unity are useful because they often allow one to extend local constructions to the whole space. For instance, the integral of differential forms on paracompact manifolds is first defined locally (where the manifold looks like Euclidean space and the integral is well known), and this definition is then extended to the whole space via a partition of unity.

Proof that paracompact Hausdorff spaces admit partitions of unity

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(Click "show" at right to see the proof or "hide" to hide it.)

A Hausdorff space is paracompact if and only if it every open cover admits a subordinate partition of unity. The if direction is straightforward. Now for the only if direction, we do this in a few stages.

Lemma 1: If is a locally finite open cover, then there exists open sets for each , such that each and is a locally finite refinement.
Lemma 2: If is a locally finite open cover, then there are continuous functions such that and such that is a continuous function which is always non-zero and finite.
Theorem: In a paracompact Hausdorff space , if is an open cover, then there exists a partition of unity subordinate to it.
Proof (Lemma 1):
Let be the collection of open sets meeting only finitely many sets in , and whose closure is contained in a set in . One can check as an exercise that this provides an open refinement, since paracompact Hausdorff spaces are regular, and since is locally finite. Now replace by a locally finite open refinement. One can easily check that each set in this refinement has the same property as that which characterised the original cover.
Now we define . The property of guarantees that every is contained in some . Therefore is an open refinement of . Since we have , this cover is immediately locally finite.
Now we want to show that each . For every , we will prove that . Since we chose to be locally finite, there is a neighbourhood of such that only finitely many sets in have non-empty intersection with , and we note those in the definition of . Therefore we can decompose in two parts: who intersect , and the rest who don't, which means that they are contained in the closed set . We now have . Since and , we have for every . And since is the complement of a neighbourhood of , is also not in . Therefore we have .
Proof (Lemma 2):
Applying Lemma 1, let be continuous maps with and (by Urysohn's lemma for disjoint closed sets in normal spaces, which a paracompact Hausdorff space is). Note by the support of a function, we here mean the points not mapping to zero (and not the closure of this set). To show that is always finite and non-zero, take , and let a neighbourhood of meeting only finitely many sets in ; thus belongs to only finitely many sets in ; thus for all but finitely many ; moreover for some , thus ; so is finite and . To establish continuity, take as before, and let , which is finite; then , which is a continuous function; hence the preimage under of a neighbourhood of will be a neighbourhood of .
Proof (Theorem):
Take a locally finite subcover of the refinement cover: . Applying Lemma 2, we obtain continuous functions with (thus the usual closed version of the support is contained in some , for each ; for which their sum constitutes a continuous function which is always finite non-zero (hence is continuous positive, finite-valued). So replacing each by , we have now — all things remaining the same — that their sum is everywhere . Finally for , letting be a neighbourhood of meeting only finitely many sets in , we have for all but finitely many since each . Thus we have a partition of unity subordinate to the original open cover.

Relationship with compactness

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There is a similarity between the definitions of compactness and paracompactness: For paracompactness, "subcover" is replaced by "open refinement" and "finite" by is replaced by "locally finite". Both of these changes are significant: if we take the definition of paracompact and change "open refinement" back to "subcover", or "locally finite" back to "finite", we end up with the compact spaces in both cases.

Paracompactness has little to do with the notion of compactness, but rather more to do with breaking up topological space entities into manageable pieces.

Comparison of properties with compactness

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Paracompactness is similar to compactness in the following respects:

It is different in these respects:

  • A paracompact subset of a Hausdorff space need not be closed. In fact, for metric spaces, all subsets are paracompact.
  • A product of paracompact spaces need not be paracompact. The square of the real line R in the lower limit topology is a classical example for this.

Variations

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There are several variations of the notion of paracompactness. To define them, we first need to extend the list of terms above:

A topological space is:

  • metacompact if every open cover has an open point-finite refinement.
  • orthocompact if every open cover has an open refinement such that the intersection of all the open sets about any point in this refinement is open.
  • fully normal if every open cover has an open star refinement, and fully T4 if it is fully normal and T1 (see separation axioms).

The adverb "countably" can be added to any of the adjectives "paracompact", "metacompact", and "fully normal" to make the requirement apply only to countable open covers.

Every paracompact space is metacompact, and every metacompact space is orthocompact.

Definition of relevant terms for the variations

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  • Given a cover and a point, the star of the point in the cover is the union of all the sets in the cover that contain the point. In symbols, the star of x in U = {Uα : α in A} is
The notation for the star is not standardised in the literature, and this is just one possibility.
  • A star refinement of a cover of a space X is a cover of the same space such that, given any point in the space, the star of the point in the new cover is a subset of some set in the old cover. In symbols, V is a star refinement of U = {Uα : α in A} if for any x in X, there exists a Uα in U such that V*(x) is contained in Uα.
  • A cover of a space X is point-finite (or point finite) if every point of the space belongs to only finitely many sets in the cover. In symbols, U is point finite if for any x in X, the set is finite.

As the names imply, a fully normal space is normal and a fully T4 space is T4. Every fully T4 space is paracompact. In fact, for Hausdorff spaces, paracompactness and full normality are equivalent. Thus, a fully T4 space is the same thing as a paracompact Hausdorff space.

Without the Hausdorff property, paracompact spaces are not necessarily fully normal. Any compact space that is not regular provides an example.

A historical note: fully normal spaces were defined before paracompact spaces, in 1940, by John W. Tukey.[12] The proof that all metrizable spaces are fully normal is easy. When it was proved by A.H. Stone that for Hausdorff spaces full normality and paracompactness are equivalent, he implicitly proved that all metrizable spaces are paracompact. Later Ernest Michael gave a direct proof of the latter fact and M.E. Rudin gave another, elementary, proof.

See also

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Notes

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Paracompact space

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In , a paracompact space is a in which every open cover admits a locally finite open refinement. This condition ensures that the space has a "tame" structure allowing for refinements that are finite in a local sense, generalizing the global finiteness of compact spaces while applying to non-compact examples like Euclidean spaces. The concept was introduced by in 1944 as a of to facilitate the study of infinite-dimensional spaces and manifolds. Paracompact spaces often incorporate the Hausdorff separation axiom in their definition to ensure desirable geometric properties, though the core covering condition can be stated more generally. Key examples include all compact Hausdorff spaces, which trivially satisfy the refinement property since any open cover has a finite subcover that is locally finite; metric spaces, which admit such refinements via the Nagata-Smirnov metrization ; and Rn\mathbb{R}^n for any nn, including infinite dimensions under suitable topologies. Disjoint unions and closed subspaces of paracompact spaces inherit the property, making it stable under common constructions in . Important theorems highlight the utility of paracompactness: every paracompact is normal, allowing continuous functions to separate disjoint closed sets, and fully normal, meaning closed subsets are also normal in the . Second-countable locally compact are paracompact, a result crucial for manifolds and providing an equivalence with certain countability and conditions in premanifold contexts. In and analysis, paracompactness enables the existence of partitions of unity subordinate to open covers, which are essential for gluing local constructions into global ones on manifolds. While every manifold is paracompact, not all topological spaces are—counterexamples include the long line, which is sequentially but fails paracompactness.

Definition and Characterizations

Formal Definition

A XX is called paracompact if every open cover of XX admits a locally finite open refinement. This definition is to Dieudonné (1944), who originally included the Hausdorff (separated space) assumption. It generalizes by replacing finite subcovers with locally finite refinements. An open cover of a XX is a collection U={Ui}i∈I\mathcal{U} = \{U_i\}_{i \in I} of open subsets of XX such that their union equals XX, i.e., ⋃i∈IUi=X\bigcup_{i \in I} U_i = X. A refinement of an open cover U\mathcal{U} is another open cover V\mathcal{V} of XX such that for each V∈VV \in \mathcal{V}, there exists some U∈UU \in \mathcal{U} with V⊆UV \subseteq U. An open cover V\mathcal{V} is locally finite if for every point x∈Xx \in X, there exists an open neighborhood WW of xx that intersects at most finitely many sets in V\mathcal{V}. For Hausdorff paracompact spaces, this condition is equivalent to the existence of a subordinate to every open cover.

Equivalent Formulations

A paracompact space possesses several equivalent characterizations under additional assumptions such as Hausdorff or regularity. One foundational formulation, introduced by in 1944 for Hausdorff spaces, states that every open cover has a star-refinement: an open refinement V\mathcal{V} such that for each V∈VV \in \mathcal{V}, the star St(V,V)=⋃{W∈V∣V∩W≠∅}\mathrm{St}(V, \mathcal{V}) = \bigcup \{ W \in \mathcal{V} \mid V \cap W \neq \emptyset \} is an . For s, paracompactness is equivalent to every open cover admitting an open point-finite refinement, meaning a refinement where each point lies in only finitely many members of the cover. To see that a locally finite open refinement implies a point-finite one, note that local finiteness directly ensures each point is covered by finitely many sets, as the finite collection around any point covers a neighborhood. Conversely, in a , a point-finite open refinement U\mathcal{U} can be refined to a locally finite one by shrinking each U∈UU \in \mathcal{U} to a FU⊂UF_U \subset U such that the stars remain covered appropriately, leveraging regularity to separate points from boundaries; the collection of interiors of these closures then forms a locally finite refinement. Paracompactness is also equivalent to the star-refinement property in s. A locally finite open refinement U\mathcal{U} yields a star-refinement by taking U\mathcal{U} itself, since the star of any U∈UU \in \mathcal{U} is covered by the finite open sets intersecting it, and regularity ensures an open neighborhood contained in the union. In the reverse direction, a star-refinement V\mathcal{V} in a allows construction of a locally finite refinement by selecting, for each V∈VV \in \mathcal{V}, an open set WV⊂St(V,V)W_V \subset \mathrm{St}(V, \mathcal{V}) whose closure is contained in St(V,V)\mathrm{St}(V, \mathcal{V}), ensuring the stars align to form a locally finite cover via finite intersections around points. In s, paracompactness is further equivalent to full normality: every open cover has a closed, locally finite refinement. This , known as Stone's theorem, holds because a paracompact allows shrinking open refinements to closed ones using Hausdorff separation, yielding closed locally finite covers; conversely, a fully normal admits partitions of unity subordinate to open covers, which refine to locally finite open covers.

Examples

Positive Examples

Compact Hausdorff spaces provide fundamental examples of paracompact spaces. In such spaces, every open cover admits a finite subcover, which serves as a locally finite open refinement, thereby satisfying the paracompact condition. Metrizable spaces, whether complete or incomplete, are also paracompact. This follows from the fact that every allows for the construction of a locally finite refinement for any open cover, as established by classical results in topology. Specific instances include Euclidean spaces Rn\mathbb{R}^n for any n∈Nn \in \mathbb{N} and nn-spheres SnS^n, both of which admit natural metrics and thus inherit paracompactness. Smooth manifolds, including both finite- and infinite-dimensional cases, are paracompact. This property is crucial in , enabling the existence of partitions of unity subordinate to any open cover, which facilitates the of smooth functions and vector fields. A notable example of a paracompact space that is not normal is Hewitt's , a compact that fails normality. This example highlights the independence of these separation properties in non-Hausdorff settings.

Counterexamples

A prominent counterexample to paracompactness is the long line, constructed as the set ω1×[0,1)\omega_1 \times [0,1) equipped with the order topology from the lexicographic order, where ω1\omega_1 denotes the first uncountable ordinal. This space is Hausdorff, connected, locally Euclidean, and countably paracompact, yet it fails to be paracompact because the open cover {[0,α)∣α<ω1}\{ [0, \alpha) \mid \alpha < \omega_1 \} admits no locally finite open refinement: any such refinement covering points arbitrarily close to ω1\omega_1 would intersect uncountably many members of the cover in every neighborhood. The open ordinal space [0,ω1)[0, \omega_1), with the order topology, provides another illustration of non-paracompactness. This space is countably compact and hereditarily normal but not paracompact, as the cover {[0,α)∣α<ω1}\{ [0, \alpha) \mid \alpha < \omega_1 \} has no locally finite refinement; neighborhoods of limit ordinals intersect infinitely many sets without a finite bound locally. Dieudonné's plank offers a more intricate counterexample, defined as the product [0,ω1]×[0,ω][0, \omega_1] \times [0, \omega] minus the point (ω1,ω)(\omega_1, \omega), using the order topology. The resulting space is (completely regular), countably compact, and metacompact but not paracompact. The specific open cover consisting of {{ω1}×[0,n)∣n<ω}∪{[0,α]×{ω}∣α<ω1}\{ \{\omega_1\} \times [0, n) \mid n < \omega \} \cup \{ [0, \alpha] \times \{\omega\} \mid \alpha < \omega_1 \} lacks a locally finite open refinement, since any refinement near the "top edge" {ω1}×[0,ω]\{ \omega_1 \} \times [0, \omega] would require intersecting uncountably many vertical strips without local finiteness. The Sorgenfrey line, the real line R\mathbb{R} with the lower limit topology generated by half-open intervals [a,b)[a, b), is paracompact and even hereditarily Lindelöf. In contrast, the Sorgenfrey plane, the product of two copies of this line, fails paracompactness: it is Hausdorff but not normal (as the rational and irrational antidiagonals cannot be separated by disjoint open sets), and since every paracompact Hausdorff space is normal, the plane cannot be paracompact. This highlights how paracompactness does not preserve under finite products. Uncountable discrete spaces, where every subset is open, actually satisfy paracompactness; the singleton cover is open and locally finite, as each point's singleton neighborhood intersects only one set. However, such spaces illustrate related failures, like non-Lindelöfness, which can interact with paracompactness in subspaces or products.

Properties

Topological Properties

A paracompact regular space is normal, meaning that for any two disjoint closed subsets AA and BB, there exist disjoint open sets UU and VV such that A⊆UA \subseteq U and B⊆VB \subseteq V. To see this, consider the open cover consisting of all open sets containing points of AA but disjoint from BB (possible by regularity) and the complement of A∪BA \cup B. By paracompactness, this cover has a locally finite open refinement {Vi}\{V_i\}. Define U=⋃{Vi:Vi∩B=∅}U = \bigcup \{V_i : V_i \cap B = \emptyset\}; then UU is open, contains AA, and is disjoint from BB since each ViV_i intersecting BB is excluded, and the local finiteness ensures no overlap issues in separation. If the space is additionally Hausdorff, then it is collectionwise normal: for any discrete collection of closed sets {Aα}\{A_\alpha\}, there exist pairwise disjoint open sets UαU_\alpha such that Aα⊆UαA_\alpha \subseteq U_\alpha for each α\alpha. This follows from constructing, via paracompactness, a locally finite refinement of the cover by sets intersecting at most one AαA_\alpha, and then using the Hausdorff property to shrink neighborhoods around each AαA_\alpha disjointly. Every paracompact Hausdorff space is thus normal, as Hausdorff implies regular in this context. A second-countable paracompact Hausdorff space is metrizable (since it is regular) and thus admits a σ\sigma-locally finite basis, by taking countable unions of balls of radius 1/n1/n around a countable dense set for n∈Nn \in \mathbb{N}. However, paracompact spaces do not generally admit bases, let alone σ\sigma-locally finite ones; for example, the Sorgenfrey line is paracompact but lacks a countable basis. Paracompactness is preserved under certain operations: every closed subspace of a paracompact space is paracompact, as open covers of the subspace lift to the ambient space and refine accordingly. Moreover, the product of a paracompact space with a compact Hausdorff space is paracompact.

Covering Refinements

In paracompact Hausdorff spaces, locally finite open refinements of any given open cover can be constructed at a high level using partitions of unity subordinated to the cover, where the supports of the continuous functions in the partition allow for the definition of smaller open sets that form a locally finite family covering the space. This approach leverages the existence of such partitions to ensure the refinement intersects each point in only finitely many sets, maintaining the topological structure without overlapping excessively. A key feature of paracompact spaces is shrinkability: for every open cover {Uα}α∈A\{U_\alpha\}_{\alpha \in A}, there exists an open refinement {Vα}α∈A\{V_\alpha\}_{\alpha \in A} such that the closure Vα‾⊂Uα\overline{V_\alpha} \subset U_\alpha for each α\alpha, ensuring the refined sets are properly contained within the original ones while preserving openness and coverage. This property arises from the ability to adjust the cover using the regularity of the space to separate closures appropriately. Paracompact spaces guarantee point-finite refinements, where each point in the space lies in only finitely many members of the refinement; this finite intersection property at each point enables applications such as barycentric coordinates, which assign weights to points based on their membership in the simplicial structure induced by the cover, facilitating coordinate representations and approximations in the topology. A fundamental theorem states that in any paracompact space, every open cover admits a closed locally finite refinement, combining closure properties with controlled finiteness to strengthen the refinement's utility in further topological constructions. In regular paracompact spaces, a specific method for obtaining refinements involves first selecting a locally finite open refinement of the original cover, then using the regularity axiom to shrink each open set to a closed subset whose interior still covers the space, yielding a closed locally finite refinement that can be further adjusted for additional properties like σ-local finiteness if needed.

Paracompact Hausdorff Spaces

Partitions of Unity

In paracompact Hausdorff spaces, partitions of unity serve as a powerful tool for constructing and gluing local data into global structures. A partition of unity subordinate to an open cover {Ui}i∈I\{U_i\}_{i \in I} of a topological space XX consists of a locally finite family of continuous functions ϕi:X→[0,1]\phi_i: X \to [0,1] such that the support of each ϕi\phi_i is contained in UiU_i and ∑i∈Iϕi(x)=1\sum_{i \in I} \phi_i(x) = 1 for every x∈Xx \in X. This family allows for the weighted averaging of local properties defined on the UiU_i, ensuring compatibility across overlaps. The existence of partitions of unity subordinate to every open cover provides an equivalent characterization of paracompactness in Hausdorff spaces. Specifically, a Hausdorff space is paracompact if and only if every open cover admits such a partition. This fundamental result, known as Stone's theorem, was established by Arthur H. Stone in 1948 and highlights the interplay between covering properties and functional constructions in topology. Partitions of unity find wide applications in differential geometry, analysis, and algebraic topology. In the context of manifolds, they enable the integration of continuous functions and differential forms by breaking down global integrals into locally computable pieces over coordinate charts, leveraging the paracompactness of the manifold to ensure local finiteness. They also support the extension of continuous (or smooth) functions from closed subsets to the entire space, generalizing through successive local extensions glued via the partition. In homotopy theory, partitions of unity on numerable covers facilitate the construction of global maps and homotopies by combining local ones, which is crucial for results like the homotopy extension property in cell complexes. The construction of such partitions in paracompact Hausdorff spaces begins with the paracompactness property, which guarantees a locally finite open refinement {Vj}j∈J\{V_j\}_{j \in J} of the original cover {Ui}\{U_i\}. For each VjV_j, one selects a shrinking open set Wj⊂VjW_j \subset V_j such that Wj‾⊂Vj\overline{W_j} \subset V_j, and then applies Urysohn's lemma—available due to the normality of paracompact Hausdorff spaces—to produce continuous functions ψj:X→[0,1]\psi_j: X \to [0,1] with supp⁡(ψj)⊂Vj\operatorname{supp}(\psi_j) \subset V_j and ψj=1\psi_j = 1 on Wj‾\overline{W_j}. Normalizing these by dividing by their sum yields the desired partition subordinate to the original cover.

Proof of Partition Existence

To prove that every paracompact Hausdorff space admits a partition of unity subordinate to any given open cover, proceed by constructing such a partition explicitly using the space's properties. Let XX be a paracompact Hausdorff space and let {Ui}i∈I\{U_i\}_{i \in I} be an arbitrary open cover of XX. Since XX is paracompact, there exists a locally finite open refinement {Vj}j∈J\{V_j\}_{j \in J} of {Ui}\{U_i\}, meaning each VjV_j is open, ⋃jVj=X\bigcup_j V_j = X, for each x∈Xx \in X there is a neighborhood of xx intersecting only finitely many VjV_j, and Vj⊆Ui(j)V_j \subseteq U_{i(j)} for some index i(j)∈Ii(j) \in I. Paracompact Hausdorff spaces are normal (and hence regular). Apply the shrinking lemma for paracompact spaces: there exists another locally finite open cover {Wj}j∈J\{W_j\}_{j \in J} such that Wj‾⊆Vj\overline{W_j} \subseteq V_j for each jj, where Wj‾\overline{W_j} denotes the closure of WjW_j in the normal topology of XX. Define Fj=Wj‾F_j = \overline{W_j} for each jj; then {Fj}j∈J\{F_j\}_{j \in J} is a locally finite family of closed sets with Fj⊆Vj⊆Ui(j)F_j \subseteq V_j \subseteq U_{i(j)} and ⋃jFj=X\bigcup_j F_j = X, since the WjW_j refine the cover and their closures remain within the VjV_j. The local finiteness of the FjF_j follows from that of the WjW_j and VjV_j, ensuring that at each point x∈Xx \in X, only finitely many FjF_j contain xx. For each j∈Jj \in J, since XX is normal, guarantees the existence of a continuous function uj:X→[0,1]u_j: X \to [0,1] such that uj∣Fj≡1u_j|_ {F_j} \equiv 1 and supp⁡(uj)⊆Vj\operatorname{supp}(u_j) \subseteq V_j, where the support is the closure of the set where uj>0u_j > 0. The Hausdorff assumption ensures the separation required for normality and thus for applying Urysohn's lemma to disjoint closed sets FjF_j and X∖VjX \setminus V_j. Define s:X→[0,∞)s: X \to [0, \infty) by s(x)=∑j∈Juj(x)s(x) = \sum_{j \in J} u_j(x). Local finiteness implies that for each x∈Xx \in X, only finitely many uj(x)u_j(x) are nonzero, so ss is a finite sum at every point and hence continuous as a sum of continuous functions. Moreover, s(x)≥1s(x) \geq 1 for all x∈Xx \in X, because x∈Fkx \in F_k for some kk with uk(x)=1u_k(x) = 1. Finally, define ϕj:X→[0,1]\phi_j: X \to [0,1] by ϕj(x)=uj(x)s(x)\phi_j(x) = \frac{u_j(x)}{s(x)} for s(x)>0s(x) > 0 (which holds everywhere), and note that local finiteness ensures the denominator is a finite sum. Each ϕj\phi_j is continuous as a quotient of continuous functions with nonzero denominator. The family {ϕj}j∈J\{\phi_j\}_{j \in J} forms a subordinate to {Ui}\{U_i\}, since ∑jϕj(x)=∑juj(x)/s(x)=s(x)/s(x)=1\sum_j \phi_j(x) = \sum_j u_j(x)/s(x) = s(x)/s(x) = 1 for all x∈Xx \in X, and supp⁡(ϕj)⊆supp⁡(uj)⊆Vj⊆Ui(j)\operatorname{supp}(\phi_j) \subseteq \operatorname{supp}(u_j) \subseteq V_j \subseteq U_{i(j)}. This construction holds without assuming second countability, relying solely on the general definition of paracompactness via arbitrary open covers and the normality derived from the Hausdorff condition.

Relation to Compactness

Shared Properties

Paracompact spaces share several key properties with , particularly in their behavior regarding open covers and subspace operations. A fundamental commonality is that every open cover of a admits a locally finite open refinement, as the finite subcover of any open cover is itself locally finite—each point has a neighborhood intersecting only finitely many sets in the subcover, since the subcover is finite overall. This directly implies the theorem that every is paracompact, establishing as a special case of paracompactness. Both classes of spaces are closed under the formation of closed subspaces: if XX is paracompact (or compact) and A⊆XA \subseteq X is closed, then AA inherits the paracompact (or compact) property in the . Compactness is preserved under finite products, and paracompactness is preserved under the product of a paracompact and a . Regarding images under continuous maps, compact spaces are closed under arbitrary continuous images, meaning the continuous image of a is compact. Paracompact spaces exhibit analogous behavior under more restrictive maps: the closed continuous image of a paracompact is again paracompact Hausdorff. In the Hausdorff setting, both and paracompact spaces are normal, allowing disjoint closed sets to be separated by disjoint open sets. Moreover, paracompact are collectionwise normal, meaning that for any discrete collection of closed sets, there exist disjoint open sets containing each; this property also holds for .

Contrasting Behaviors

While compactness requires that every open cover admits a finite subcover, paracompactness weakens this condition by requiring only a locally finite open refinement for every open cover. This distinction allows paracompact spaces to handle infinite covers in a controlled manner, where each point has a neighborhood intersecting only finitely many sets in the refinement, but without the global finiteness enforced by . A classic non-compact paracompact is the real line R\mathbb{R}, which admits open covers without finite subcovers—such as the cover {(n−1,n+1)∣n∈Z}\{(n-1, n+1) \mid n \in \mathbb{Z}\}—yet every open cover of R\mathbb{R} has a locally finite open refinement due to its metrizability. In contrast, compact spaces like the unit interval [0,1][0,1] always yield finite subcovers from any such cover. In paracompact but non-compact spaces, the Heine-Borel theorem fails: closed and bounded sets need not be compact. For instance, in the infinite-dimensional ℓ2\ell^2, the closed unit ball {x∈ℓ2∣∥x∥≤1}\{x \in \ell^2 \mid \|x\| \leq 1\} is closed and bounded but not compact, as it contains no finite ϵ\epsilon-net for ϵ<1\epsilon < 1 by Riesz's lemma; ℓ2\ell^2 itself is paracompact as a . Similar failures occur in paracompact manifolds of infinite dimension, where bounded closed subsets evade compactness. Paracompactness does not generally imply sequential compactness or limit point compactness. In R\mathbb{R}, which is paracompact, the sequence (n)n∈N(n)_{n \in \mathbb{N}} has no convergent subsequence, so R\mathbb{R} is neither sequentially compact nor limit point compact (as the set N\mathbb{N} has no limit point). Conversely, sequential compactness does not imply paracompactness, inverting the situation for compact spaces. The long line, constructed as [0,ω1)×[0,1)[0, \omega_1) \times [0,1) with the order topology (where ω1\omega_1 is the first uncountable ordinal), is sequentially compact but not paracompact, as its standard cover by countable intervals lacks a locally finite refinement.

Variations

Countable Paracompactness

A topological space is countably paracompact if every countable open cover admits a locally finite open refinement. This condition represents a weakening of full paracompactness, which requires such a refinement for every open cover, not just countable ones. Countably paracompact spaces form an intermediate class between compact spaces and more general topological spaces, capturing behaviors relevant to covering properties in dimensions where uncountable covers are not immediately necessary. The concept was introduced by C. H. Dowker in 1951 as part of investigations into refinements and their implications for normality and product spaces. Dowker's work built on earlier ideas of paracompactness from Dieudonné (1944) and explored how countable restrictions simplify proofs in and metrization theory. A key property is that every countably paracompact has the feature that its product with the unit interval [0,1] is normal; conversely, if a has this product property, it is countably paracompact. This characterization parallels the full paracompact case but applies only to countable covers. Additionally, the product of a countably paracompact space with any remains countably paracompact. Perfectly normal spaces—those that are normal and in which every is a G_δ set—are always countably paracompact. Countably compact spaces are countably paracompact, as any countable open cover has a finite subcover, which is locally finite. The space of countable ordinals ω₁ with the provides a classic example: it is countably compact (hence countably paracompact) but fails paracompactness due to the uncountable open cover consisting of initial segments [0,α) for successor ordinals α < ω₁, which admits no locally finite refinement. The Dieudonné plank, defined as ([0,ω₁] × [0,ω]) \ {(ω₁, ω)}, is another example: it is normal and countably paracompact but not paracompact, as certain uncountable covers lack locally finite refinements. In spaces with a countable basis, such as , countable paracompactness is equivalent to full paracompactness. This follows because every open cover in a has a countable subcover; thus, ensuring locally finite refinements for countable covers suffices for all covers. Countably paracompactness is not preserved under arbitrary countable products, but specific cases like products with compact factors maintain the property.

Other Generalizations

A XX is said to be paracompact at infinity if its Stone-Čech βX∖X\beta X \setminus X is paracompact. This notion is particularly relevant for studying s in compactifications of non-locally compact spaces, where it ensures that the "behavior at infinity" mirrors standard paracompactness properties. Subparacompactness provides a weakening of paracompactness, defined as the property that every open cover of the space admits a point-finite closed refinement. This condition is distinct from metacompactness, which requires a point-finite open refinement, and the two properties are incomparable—there exist metacompact spaces that are not subparacompact. Subparacompactness relates to structures through its implications for σ-discrete refinements in normal spaces. Subparacompact spaces are preserved under countable products in certain contexts, such as when combined with regularity assumptions. Moore spaces generalize paracompactness in the context of developable spaces: a Moore space is a completely regular space admitting a development, which is a of open covers such that for each point and neighborhood, all but finitely many covers refine appropriately around that point. If a Moore space is metrizable, it is necessarily paracompact, as metrizability implies the existence of locally finite refinements for any open cover. However, non-metrizable Moore spaces may fail to be paracompact, highlighting the role of metrizability in bridging development and refinement properties. Ultraparacompactness strengthens paracompactness further, requiring that every open cover admits an open refinement that partitions the space into disjoint sets. Equivalently, such a refinement consists of clopen sets, making ultraparacompact spaces hereditarily zero-dimensional when combined with other separation axioms. This property is exemplified by discrete spaces and certain tree topologies, where it facilitates strong control over covers via partitions. In , generalizations of paracompactness extend to sheaf theory and étale spaces, where refinement properties ensure compatibility between Čech and sheaf . For instance, on paracompact bases, every sheaf admits fine resolutions, enabling the computation of via partitions of unity subordinate to covers. In the étale topology, analogous conditions on sites replace open covers with étale morphisms, allowing sheaf-theoretic generalizations that capture descent data beyond classical paracompactness. These extensions, while rooted in refinement ideals, adapt to algebraic stacks and rigid analytic spaces, where traditional topological paracompactness may not directly apply.

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