Paracompact Spaces

Etymology

The notion of paracompactness was introduced in 1944 by Jean Dieudonné in his article ‘Une généralisation des espaces compacts’ published in Journal de mathématiques pures et appliquées.

The title of the original article reflects itself in the choice of the term paracompact, as the prefix ‘para’ coming from the greek ‘παρά’ means ‘beside’.

Definition

A topological space is said to be paracompact if it is Hausdorff and if every open cover admits a refinement that is locally finite.

Generalisation of the notion of compactness

Let us first recall the — french — definition of a compact topological space: ‘A topological space is said to be compact if it is Hausdorff and if every open cover admits a finite subcover.’

Because of the finiteness condition, this definition can also be rewritten in the following way: ‘A topological space is said to be compact if it is Hausdorff and if every open cover admits a finite refinement’.

That way, paracompactness appears to be the ‘locally finite’ version of the notion of ‘compactness’ that could be seen as the ‘globally finite’ version.

Examples

Most commonly encountered topological spaces are paracompact, which makes it a pervasive notion.

Heredity along constructions

One drawback of the notion of paracompactness is that it is not as robust as the notion of compactness along usual constructions. It is also less preserved than regularity and Lindelöfness.

In practice

Compact spaces are especially nice in geometry when you have functions going out of it. One usually like to construct and study functions — vector fields, forms etc — on a compact manifold.

It happens that paracompact spaces are nice because the theory of functions on paracompact spaces is rich. This is a consequence of the existence of partitions of unity.

Metrisation theorem

All metric spaces are paracompact. Conversely, by a theorem of Smirnov, a paracompact space is metrisable if it is locally metrisable. As a consequence all topological manifolds — which are always supposed to be paracompact, even when non explicitly stated — as metrisable.

Partitions of unity

A topological space is paracompact if and only if it admits a partition of unity. Partitions of unity allows us to construct functions locally. This is the another reason why all smooth manifolds are at least supposed to be paracompact.

Paracompact vs. regular Lindelöf

As stated above, every regular Lindelöf space is paracompact. The converse is false in general.

To get an idea of the size of the gap between the two set of notions, we shall have a look to the case of topological spaces that are locally equivalent to an Euclidian space.

A topological manifold is a paracompact space with that property. It is equivalent to saying that the topological space is metrisable.

Every topological manifold is regular but not necessarily Lindelöf. In this case the Lindelöf condition is equivalent to the fact that the topological manifold may be embedded into a finite dimensional Euclidian space. More precisely, the Lindelöf condition ensures that the paracompact manifold has only a countable number of connected components. Hence every connected manifold is Lindelöf.

Riemann surfaces

Suppose we have a Hausdorff topological space with a holomorphic and one dimentional atlas, then surprisingly, it has to be paracompact [1]. Hence, there is no complex analogue of the long line.

Such a property is very specific to complex dimension one.

Computation of cohomology

On a paracompact space, sheaf cohomology — abelian or non-abelian — coincides with Eilenberg-Maclane cohomology [2].

References