Editing Paracompact space (section)

== 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:

* '''[[metacompact space|metacompact]]''' if every open cover has an open point-finite refinement.
* '''[[orthocompact space|orthocompact]]''' if every open cover has an open refinement such that the intersection of all the open sets about any point in this refinement is open.
* '''fully normal''' if every open cover has an open [[star refinement]], and '''fully T<sub>4</sub>''' if it is fully normal and [[T1 space|T<sub>1</sub>]] (see [[separation axioms]]).

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 ===

* 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''<sub>α</sub> : α in ''A''} is

: <math>\mathbf{U}^{*}(x) := \bigcup_{U_{\alpha} \ni x}U_{\alpha}.</math>

: 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 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''<sub>α</sub> : α in ''A''} if for any ''x'' in ''X'', there exists a ''U''<sub>α</sub> in ''U'' such that '''V'''<sup>*</sup>(''x'') is contained in ''U''<sub>α</sub>.
* A cover of a space ''X'' is ''[[point-finite collection|point-finite]]'' (or ''point finite'') if every point of the space belongs to only finitely many sets in the cover. In symbols, '''U''' is point finite if for any ''x'' in ''X'', the set <math>\left\{\alpha \in A : x \in U_{\alpha} \right\}</math> is finite.

As the names imply, a fully normal space is [[normal space|normal]] and a fully T<sub>4</sub> space is T<sub>4</sub>. Every fully T<sub>4</sub> space is paracompact. In fact, for Hausdorff spaces, paracompactness and full normality are equivalent. Thus, a fully T<sub>4</sub> 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]].<ref>{{cite book | last1=Tukey | first1=John W. | author1-link=John Tukey | title=Convergence and Uniformity in Topology | mr=0002515 | year=1940 | publisher=Princeton University Press, Princeton, N. J.|series= Annals of Mathematics Studies | volume=2 | pages=ix+90}}</ref> 
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
[[Mary Ellen Rudin|M.E. Rudin]] gave another, elementary, proof.