Editing Paracompact space (section)

== Examples ==

* Every [[compact space]] is paracompact.
* Every [[regular space|regular]] [[Lindelöf space]] is paracompact, by [[Michael's theorem on paracompact spaces|Michael's theorem]] in the Hausdorff case.<ref>It is not hard to give a direct proof that does not use Hausdorff.</ref> In particular, every [[locally compact]] [[Hausdorff space|Hausdorff]] [[second-countable space]] is paracompact.
* The [[Sorgenfrey line]] is paracompact, even though it is neither compact, locally compact, second countable, nor metrizable.
* Every [[CW complex]] is paracompact.<ref>[[Allen Hatcher|Hatcher, Allen]], ''Vector bundles and K-theory'', preliminary version available on the [http://pi.math.cornell.edu/~hatcher/ author's homepage]</ref>
* ('''Theorem of [[A. H. Stone]]''') Every [[metric space]] is paracompact.<ref>Stone, A. H.  [https://www.ams.org/journals/bull/1948-54-10/S0002-9904-1948-09118-2 Paracompactness and product spaces].  Bull. Amer. Math. Soc. 54 (1948), 977–982</ref>  Early proofs were somewhat involved, but an elementary one was found by [[Mary Ellen Rudin|M.&nbsp;E.&nbsp;Rudin]].<ref>{{cite journal
| last1=Rudin | first1=Mary Ellen | authorlink1=Mary Ellen Rudin
| title=A new proof that metric spaces are paracompact
| journal=[[Proceedings of the American Mathematical Society]]
| volume=20
| issue=2
| date=February 1969
| pages=603
| doi=10.1090/S0002-9939-1969-0236876-3 | doi-access=free}}</ref>  Existing proofs of this require the [[axiom of choice]] for the non-[[separable space|separable]] case.  It has been shown that [[Zermelo–Fraenkel set theory|ZF]] theory is not sufficient to prove it, even after the weaker [[axiom of dependent choice]] is added.<ref>{{cite journal
| last1=Good | first1=C.
| last2=Tree | first2=I. J.
| last3=Watson | first3=W. S.
| title=On Stone's theorem and the axiom of choice
| journal=[[Proceedings of the American Mathematical Society]]
| volume=126
| issue=4
| date=April 1998
| pages=1211–1218
| doi=10.1090/S0002-9939-98-04163-X | doi-access=free}}</ref>
*A Hausdorff space admitting an [[exhaustion by compact sets]] is paracompact.

Some examples of spaces that are not paracompact include:

* The most famous counterexample is the [[long line (topology)|long line]], which is a nonparacompact [[topological manifold]]. (The long line is locally compact, but not second countable.)
* Another counterexample is a [[product topology|product]] of [[uncountable set|uncountably]] many copies of an [[infinite (cardinality)|infinite]] [[discrete space]]. Any infinite set carrying the [[particular point topology]] is not paracompact; in fact it is not even [[metacompact]].
* The [[Prüfer manifold]] ''P'' is a non-paracompact surface. (It is easy to find an uncountable open cover of ''P'' with no refinement of any kind.)
* The [[bagpipe theorem]] shows that there are 2<sup>ℵ<sub>1</sub></sup> topological [[Equivalence class|equivalence classes]] of non-paracompact surfaces.
* The [[Sorgenfrey plane]] is not paracompact despite being a product of two paracompact spaces.