In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a paracompact space is a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 in which every open cover admits a locally finite

Locally finite collection

In the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space. It is fundamental in the study of paracompactness and topological dimension....

 open refinement. Paracompact spaces are sometimes also required to be Hausdorff

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...

. Paracompact spaces were introduced by .

Definitions of relevant terms

• A cover of a set X is a collection of subset
  Subset
  In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
  
  s of X whose union
  Union (set theory)
  In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
  
   is X. In symbols, if U = {U_α : α in A} is an indexed family of subsets of X, then U is a cover if and only if
  If and only if
  In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
  
  

• A cover of a topological space X is open if all its members are open set
  Open set
  The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
  
  s. In symbols, a cover U is an open cover if U is a subset of T, where T is the topology on X.
• A refinement of a cover of a space X is a new cover of the same space such that every set in the new cover is a subset
  Subset
  In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
  
   of some set in the old cover. In symbols, the cover V = {V_β : β in B} is a refinement of the cover U = {U_α : α in A} if and only if, for any
  Universal quantification
  In predicate logic, universal quantification formalizes the notion that something is true for everything, or every relevant thing....
  
   V_β in V, there exists some
  Existential quantification
  In predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain. It is denoted by the logical operator symbol ∃ , which is called the existential quantifier...
  
   U_α in U such that V_β is contained in U_α.
• An open cover of a space X is locally finite if every point of the space has a neighborhood which intersects only finitely many sets in the cover. In symbols, U = {U_α : α in A} is locally finite if and only if, for any x in X, there exists some neighbourhood V(x) of x such that the set

is finite.

• A topological space X is called paracompact if any open cover of X admits an open refinement that is locally finite.

Note the similarity between the definitions of compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 and paracompact: for paracompact, we replace "subcover" by "open refinement" and "finite" by "locally finite". Both of these changes are significant: if we take the above 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.

A hereditarily paracompact space is a space such that every subspace of it is paracompact. This is equivalent to requiring that every open subspace be paracompact.

Examples and counterexamples

• Every compact space
  Compact space
  In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
  
   is paracompact.
• (Theorem of A. H. Stone) Every metric space
  Metric space
  In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
  
   is paracompact. Early proofs were somewhat involved, but an elementary one was found by M. E. Rudin
  Mary Ellen Rudin
  Mary Ellen Rudin is an American mathematician.Born Mary Ellen Estill, she attended the University of Texas, completing her B.A. in 1944 and her Ph.D. in 1949, under Robert Lee Moore. In 1953, she married the mathematician Walter Rudin. Following her mentor Moore, her research centers on point-set...
  
  . Existing proofs of this require the axiom of choice for the non-separable case. It has been shown that neither ZF theory
  Zermelo–Fraenkel set theory
  In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...
  
   nor ZF theory with the axiom of dependent choice
  Axiom of dependent choice
  In mathematics, the axiom of dependent choices, denoted DC, is a weak form of the axiom of choice which is still sufficient to develop most of real analysis...
  
   is sufficient.
• Every regular
  Regular space
  In topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods. Thus p and C...
  
   Lindelöf space
  Lindelöf space
  In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover....
  
   is paracompact. In particular, every locally compact Hausdorff
  Hausdorff space
  In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
  
   second-countable space
  Second-countable space
  In topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base...
  
   is paracompact.
• The Sorgenfrey line is paracompact, even though it is neither compact, locally compact, second countable, nor metrizable.
• Every CW complex
  CW complex
  In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
  
   is paracompact

Some examples of spaces that are not paracompact include:

• The most famous counterexample is the long line
  Long line (topology)
  In topology, the long line is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties. Therefore it serves as one of the basic counterexamples of topology...
  
  , which is a nonparacompact topological manifold
  Topological manifold
  In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
  
  . (The long line is locally compact, but not second countable.)
• Another counterexample is a product
  Product topology
  In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
  
   of uncountably
  Uncountable set
  In mathematics, an uncountable set is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.-Characterizations:There...
  
   many copies of an infinite discrete space
  Discrete space
  In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
  
  . Any infinite set carrying the particular point topology
  Particular point topology
  In mathematics, the particular point topology is a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space. Formally, let X be any set and p ∈ X. The collectionof subsets of X is then the particular point topology...
  
   is not paracompact; in fact it is not even metacompact.
• The Prüfer manifold
  Prüfer manifold
  In mathematics, the Prüfer manifold or Prüfer surface is a 2-dimensional Hausdorff real analytic manifold that is not paracompact. It was introduced by and named after Heinz Prüfer.-Construction:The Prüfer manifold can be constructed as follows...
  
   is a non-paracompact surface.

Properties

• (Theorem of Jean Dieudonné
  Jean Dieudonné
  Jean Alexandre Eugène Dieudonné was a French mathematician, notable for research in abstract algebra and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a historian of...
  
  ) Every paracompact Hausdorff space is normal
  Normal space
  In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...
  
  .
• Paracompactness is weakly hereditary, i.e. every closed subspace of a paracompact space is paracompact. This can be extended to F-sigma
  F-sigma set
  In mathematics, an Fσ set is a countable union of closed sets. The notation originated in France with F for fermé and σ for somme ....
  
   subspaces as well.
• If every open subset of a space is paracompact, then it is hereditarily paracompact.
• A regular space
  Regular space
  In topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods. Thus p and C...
  
   is paracompact if every open cover admits a locally finite refinement. (Here, the refinement is not required to be open.) In particular, every regular Lindelof space
  Lindelöf space
  In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover....
  
   is paracompact.
• (Smirnov metrization theorem) A topological space is metrizable if and only if it is paracompact, Hausdorff, and locally metrizable.
• Every paracompact Hausdorff space is a shrinking space
  Shrinking space
  In mathematics, in the field of topology, a topological space is said to be a shrinking space if every open cover admits a shrinking. A shrinking of an open cover is another open cover indexed by the same indexing set, with the property that the closure of each open set in the shrinking lies inside...
  
  , 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, the cohomology of a sheaf is equal to its Čech cohomology
  Cech cohomology
  In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech.-Motivation:...
  
  .
• Michael selection theorem
  Michael selection theorem
  In functional analysis, a branch of mathematics, the most popular version of the Michael selection theorem, named after Ernest Michael, states the following:- Other selection theorems :* Zero-dimensional Michael selection theorem...
  
   states that lower semicontinuous multifunctions from X into nonempty closed convex subsets of Banach spaces admit continuous selection iff X is paracompact.

Partitions of unity

Paracompactness has little to do with the notion of compactness, but rather more to do with breaking up topological space entities into manageable pieces. The most important feature of paracompact Hausdorff space

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...

s is that they are normal

Normal space

In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...

 and 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

Unit interval

In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...

 [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
  Support (mathematics)
  In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...
  
   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 Hausdorff space is 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 form

Differential form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...

s on paracompact manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

s is first defined locally (where the manifold looks like Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 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

An 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 refinitement. One can easily check that each set in this refinement has the same property as that which characterised the original cover.

Now we define . We have that each ; for otherwise letting , we take with closure contained in ; but then a contradiction. And it easy to see that is an open refinement of .

Finally, to verify that this cover is locally finite, fix ; let a neighbourhood of meeting only finitely many sets in . We will show that meets only finitely many of the . If meets , then some with meets . Thus is the same as which is contained in . By the setup of , each meets only finitely many sets in . Hence the right-hand collection is a finite union of finite sets. Thus is finite. Hence the cover is locally finite.

Proof (Lemma 2): Applying Lemma 1, let be coninuous 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.

Comparison with compactness

Paracompactness is similar to compactness in the following respects:

• Every closed subset of a paracompact space is paracompact.
• Every paracompact Hausdorff space is normal.

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.

Product related properties

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

• The product of a paracompact space and a compact space
  Compact space
  In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
  
   is paracompact.
• The product of a metacompact space
  Metacompact space
  In mathematics, in the field of general topology, a topological space is said to be metacompact if every open cover has a point finite open refinement...
  
   and a compact space is metacompact.

Both these results can be proved by the tube lemma

Tube lemma

In mathematics, particularly topology, the tube lemma is a useful tool in order to prove that the finite product of compact spaces is compact. It is in general, a concept of point-set topology.-Tube lemma:...

 which is used in the proof that a product of finitely many compact spaces is compact.

Variations

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

• 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 = {U_α : α in A} is

The notation for the star is not standardised in the literature, and this is just one possibility.

• A star refinement
  Star refinement
  In mathematics, specifically in the study of topology and open covers of a topological space X, a star refinement is a particular kind of refinement of an open cover of X....
  
  of a cover of a space X is a new 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 = {U_α : α in A} if and only if, for any x in X, there exists a U_α in U, such that V^*(x) is contained in U_α.
• A cover of a space X is pointwise finite if every point of the space belongs to only finitely many sets in the cover. In symbols, U is pointwise finite if and only if, for any x in X, the set

is finite.

A topological space is:

• metacompact
  Metacompact space
  In mathematics, in the field of general topology, a topological space is said to be metacompact if every open cover has a point finite open refinement...
  
  if every open cover has an open pointwise finite refinement.
• orthocompact
  Orthocompact space
  In mathematics, in the field of general topology, a topological space is said to be orthocompact if every open cover has an interior preserving open refinement...
  
  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
  T1 space
  In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...
  
   (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.

As the name implies, a fully normal space is normal

Normal space

In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...

. 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.

As an historical note: fully normal spaces were defined before paracompact spaces.
The proof that all metrizable spaces are fully normal is easy. When it was proved by A.H. Stone that for Hausdorff spaces fully normal and paracompact are equivalent, he implicitly proved that all metrizable spaces are paracompact. Later M.E. Rudin

Mary Ellen Rudin

Mary Ellen Rudin is an American mathematician.Born Mary Ellen Estill, she attended the University of Texas, completing her B.A. in 1944 and her Ph.D. in 1949, under Robert Lee Moore. In 1953, she married the mathematician Walter Rudin. Following her mentor Moore, her research centers on point-set...

gave a direct proof of the latter fact.