Paracompact space

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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.)

Table of contents

1 Definitions of relevant terms
2 Examples
3 Properties
4 Partitions of unity
5 Counterexamples
6 Variations

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 iff

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 contained in 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} iff, 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 iff, 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.

Examples

Every compact space is paracompact.
Every metric space (hence, every metrisable space) is paracompact.
Every locally compact second countable space is paracompact.
Every regular Lindelof space is paracompact.
The lower limit topology on the real line is paracompact, even though it is neither compact, locally compact, second countable, nor metrisable.

Properties

Every paracompact Hausdorff space is normal.

Paracompactness is weakly hereditary, i.e. every closed subspace of a paracompact space is paracompact.

A regular space is paracompact if every open cover admits a locally finite refinement. (Here, the refinement is not required to be open.)
(Smirnov metrisation theorem) A topological space is metrisable if and only if it is paracompact, Hausdorff, and locally metrisable.

Partitions of unity

The most important feature of paracompact Hausdorff spaces is that they are normal and admit partitions of unity relative to any open cover. This means the following: if X is a paracompact Hausdoff 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 f is identically 0 outside of 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.

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.

Counterexamples

As you might guess from the generality of most of the examples above, it's actually harder to think of spaces that aren't 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 as in the previous section, while excluding the long line, which is useless in almost every application.

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} iff, 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 iff, for any x in X, the set

is finite.

A topological space X is metacompact if every open cover has an open pointwise finite refinement, and 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.

As you might guess from the terminology, a fully normal space is normal. Any space that is fully normal must be paracompact, and any space that is paracompact must be metacompact. 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.