Information about Paracompact

In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. (Paracompact spaces are often required to be Hausdorff, but we will not make that assumption in this article.)

Definitions of relevant terms

• A cover of a set X is a collection of subsets of X whose union 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

• A cover of a topological space X is open if all its members are open sets. 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 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 V_β in V, there exists some 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.

Note the similarity between the definitions of compact 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 is paracompact.
• (Theorem of A. H. Stone) Every metric space (hence, every metrisable space) is paracompact. Early proofs were somewhat involved, but an elementary one was found by M. E. Rudin^[1] Existing proofs of this requires the axiom of choice for the non-separable case. It has been shown that neither ZF theory nor ZF theory with the principle of dependent choice is sufficient.^[2]
• Every locally compact second-countable space is paracompact.
• Every regular Lindelöf space is paracompact.
• The real line in the lower limit topology is paracompact, even though it is neither compact, locally compact, second countable, nor metrisable.

As you might guess from the generality of most of the examples above, it is actually harder to think of spaces that are not paracompact than to think of spaces that are. The most famous counterexample is the long line, which is a nonparacompact topological manifold. (The long line is locally compact, but not second countable.) Another counterexample is a product of uncountably many copies of an infinite discrete space.

Most mathematicians who use point set topology, rather than investigate it in its own right, regard nonparacompact spaces as pathological. For example, manifolds are usually defined to be paracompact, thus allowing integration of differential forms to be defined, while excluding the long line, which is useless in almost every application.

Properties

• (Theorem of Jean Dieudonné) Every paracompact Hausdorff space is normal.
• 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.
• If every open subset of a space is paracompact, then it is hereditarily paracompact.
• A regular space 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 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, 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.

Partitions of unity

The most important feature of paracompact Hausdorff spaces is that they are normal 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 [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 Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. 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.

Variations

There are several mild 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 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 if every open cover has an open pointwise 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.

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.

A fully normal space is normal. Any space that is fully normal must be paracompact, any paracompact space must be metacompact; any metacompact space must be orthocompact. In fact, for Hausdorff spaces, paracompactness and full normality are equivalent. Thus, a fully T₄ space (that is, a fully normal space that is also T₁; see Separation axioms) 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 metrisable 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 metrisable spaces are paracompact. Later M.E. Rudin gave a direct proof of the latter fact.

Similarities 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 is paracompact.
• The product of a metacompact space and a compact space is metacompact.
• The product of an orthocompact space and a compact space is orthocompact.

All 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.

References

1. General Topology by Stephen Willard, Addison-Wesley Publishing Company, 1968.