Paracompact space

In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff.
Every closed subspace of a paracompact space is paracompact. While compact subsets of Hausdorff spaces are always closed, this is not true for paracompact subsets. A space such that every subspace of it is a paracompact space is called hereditarily paracompact. This is equivalent to requiring that every open subspace be paracompact.
Tychonoff's theorem (which states that the product of any collection of compact topological spaces is compact) does not generalize to paracompact spaces in that the product of paracompact spaces need not be paracompact. However, the product of a paracompact space and a compact space is always paracompact.
Every metric space is paracompact. A topological space is metrizable if and only if it is a paracompact and locally metrizable Hausdorff space.
Definition.
A "cover" of a set formula_1 is a collection of subsets of formula_1 whose union contains formula_1. In symbols, if formula_4 is an indexed family of subsets of formula_1, then formula_6 is a cover of formula_1 if
A cover of a topological space formula_1 is "open" if all its members are open sets. A "refinement" of a cover of a space formula_1 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 formula_11 is a refinement of the cover formula_4 if and only if, for any formula_13 in formula_14, there exists some formula_15 in formula_6 such that formula_17.
An open cover of a space formula_1 is "locally finite" if every point of the space has a neighborhood that intersects only finitely many sets in the cover. In symbols, formula_4 is locally finite if and only if, for any formula_20 in formula_1, there exists some neighbourhood formula_22 of formula_20 such that the set
is finite. A topological space formula_1 is now said to be paracompact if every open cover has a locally finite open refinement.
Examples.
Some examples of spaces that are not paracompact include:
Properties.
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.
Although a product of paracompact spaces need not be paracompact, the following are true:
Both 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.
Paracompact Hausdorff spaces.
Paracompact spaces are sometimes required to also be Hausdorff to extend their properties.
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:
In fact, a T1 space is Hausdorff and 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 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.
Proof that paracompact Hausdorff spaces admit partitions of unity.
A Hausdorff space formula_26 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.
Relationship with compactness.
There is a similarity between the definitions of compactness and paracompactness:
For paracompactness, "subcover" is replaced by "open refinement" and "finite" by is replaced by "locally finite". Both of these changes are significant: if we take the 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.
Paracompactness has little to do with the notion of compactness, but rather more to do with breaking up topological space entities into manageable pieces.
Comparison of properties with compactness.
Paracompactness is similar to compactness in the following respects:
It is different in these respects:
Variations.
There are several variations of the notion of paracompactness. To define them, we first need to extend the list of terms above:
A topological space is:
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.
Definition of relevant terms for the variations.
As the name implies, a fully normal space is normal. 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.
Without the Hausdorff property, paracompact spaces are not necessarily fully normal. Any compact space that is not regular provides an example.
A historical note: fully normal spaces were defined before paracompact spaces, in 1940, by John W. Tukey. 
The proof that all metrizable spaces are fully normal is easy. When it was proved by A.H. Stone that for Hausdorff spaces full normality and paracompactness are equivalent, he implicitly proved that all metrizable spaces are paracompact. Later Ernest Michael 
gave a direct proof of the latter fact and
M.E. Rudin gave another, elementary, proof.