Paracompact space

A topological space is called paracompact if every open cover admits an open locally finite refinement. (Sometimes paracompact spaces are required to be Hausdorff, but not in Wikipedia). The terms are defined as follows:

A cover of a set X is a collection of subsets of X whose union is X. In symbols, U ⊆ P(X), where P(var>X) is the power set of X, is an open cover if ∪_U∈U U = X.
A cover of a topological space X is called open if all its members are open sets. In symbols, U ⊆ T, where T is the collection of open sets in X (the topology), is an open cover if ∪_U∈U U = X.
A refinement of a cover of X is a new cover of X such that every set in the new cover is a subset of some set in the old cover. In symbols, V is a refinement of U if, for any V ∈ V there exists some U ∈ U such that V ⊆ U.
An open cover is locally finite if every point of the space has a neighborhood which intersects only finitely many sets in the cover. In symbols, U is locally finite if, for any x ∈ X there exists some neighbourhood V of x such that the set {U ∈ U : U ∩ V ≠ ∅} is finite.

Examples

Every compact space is paracompact;
Every locally compact second countable space is paracompact;
Every metric space (or metrisable 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

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 the examples above, it's actually harder to think of spaces that aren't paracompact than to think of space 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. In fact, manifolds are often (but not in Wikipedia) defined to be paracompact, thus excluding the long line.

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 is U^*(x) := ∪_x∈U∈U U.
A star refinement of a cover is a new cover 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 if, for any x ∈ X, for some U ∈ U, V^*(x) ⊆ U.
A cover 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, for any x ∈ X, the set {U ∈ U : x ∈ U} 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.

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.