In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite.
[1] Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if[2] and only if it admits partitions of unity subordinate to any open cover.
Sometimes paracompact spaces are defined so as to always be Hausdorff.
This is equivalent to requiring that every open subspace be paracompact.
The notion of paracompact space is also studied in pointless topology, where it is more well-behaved.
[3][4] Compare this to Tychonoff's theorem, which states that the product of any collection of compact topological spaces is compact.
is locally finite iff the set of opens
[10] 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 spaces are sometimes required to also be Hausdorff to extend their properties.
The most important feature of paracompact Hausdorff spaces is that they 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.
is paracompact if and only if it every open cover admits a subordinate partition of unity.
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.
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.
As the names imply, a fully normal space is normal and a fully T4 space is T4.
In fact, for Hausdorff spaces, paracompactness and full normality are equivalent.
Without the Hausdorff property, paracompact spaces are not necessarily fully normal.
Any compact space that is not regular provides an example.
[12] 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.