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| In [[mathematics]], more specifically [[point-set topology]], a '''Moore space''' is a [[developable space|developable]] [[regular Hausdorff space]]. Equivalently, a [[topological space]] ''X'' is a Moore space if the following conditions hold:
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| * Any two distinct points can be [[separated by neighbourhoods]], and any [[closed set]] and any point in its [[Complement (set theory)|complement]] can be separated by neighbourhoods. (''X'' is a [[regular Hausdorff space]].)
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| * There is a [[countable set|countable]] collection of [[open cover]]s of ''X'', such that for any closed set ''C'' and any point ''p'' in its complement there exists a cover in the collection such that every neighbourhood of ''p'' in the cover is [[disjoint sets|disjoint]] from ''C''. (''X'' is a [[developable space]].)
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| Moore spaces are generally interesting in mathematics because they may be applied to prove interesting [[metrization theorem]]s. The concept of a Moore space was formulated by [[R. L. Moore]] in the earlier part of the 20th century.
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| ==Examples and properties==
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| #Every [[metrizable space]], ''X'', is a Moore space. If {''A''<sup>(''n'')</sup><sub>''x''</sub>} is the open cover of ''X'' (indexed by ''x'' in ''X'') by all balls of radius 1/''n'', then the collection of all such open covers as ''n'' varies over the positive integers is a development of ''X''. Since all metrizable spaces are normal, all metric spaces are Moore spaces.
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| #Moore spaces are a lot like regular spaces and different to [[normal space]]s in the sense that every subspace of a Moore space is also a Moore space.
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| #The image of a Moore space under an injective, continuous open map is always a Moore space. Note also that the image of a regular space under an injective, continuous open map is always regular.
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| #Both examples 2 and 3 suggest that Moore spaces are a lot similar to regular spaces.
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| #Neither the [[Sorgenfrey line]] nor the [[Sorgenfrey plane]] are Moore spaces because they are normal and not second countable.
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| #The [[Moore plane]] (also known as the Niemytski space) is an example of a non-metrizable Moore space.
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| #Every [[metacompact]], [[separable space|separable]], normal Moore space is metrizable. This theorem is known as Traylor’s theorem.
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| #Every [[Locally compact space|locally compact]], [[Locally connected|locally connected space]], normal Moore space is metrizable. This theorem was proved by Reed and Zenor.
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| #If <math>2^{\aleph_0}<2^{\aleph_1}</math>, then every [[separable space|separable]] [[normal space|normal]] Moore space is [[metrizable]]. This theorem is known as Jones’ theorem.
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| ==Normal Moore space conjecture==
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| For a long time, topologists were trying to prove the so-called normal Moore space conjecture: every normal Moore space is [[metrizable]]. This was inspired by the fact that all known Moore spaces that were not metrizable were also not normal. This would have been a nice [[Metrization theorems|metrization theorem]]. There were some nice partial results at first; namely properties 7, 8 and 9 as given in the previous section.
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| Here we see that we drop metacompactness from Traylor's theorem, but at the cost of a set-theoretic assumption. Another example of this is [[Fleissner's theorem]] that the [[V=L|axiom of constructibility]] implies that locally compact, normal Moore spaces are metrizable.
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| On the other hand, under the [[Continuum hypothesis]] (CH) and also under [[Martin's Axiom]] and not CH, there are several examples of non-metrizable normal Moore spaces. Nyikos proved that, under the so-called PMEA (Product Measure Extension Axiom), which needs a [[Large cardinal axiom|large cardinal]], all normal Moore spaces are metrizable. Finally, it was shown later that any model of ZFC in which the conjecture holds, implies the existence of a model with a large cardinal. So large cardinals are needed essentially.
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| [[Robert Lee Moore|Moore]] himself proved the theorem that a [[collectionwise normal]] Moore space is metrizable, so strengthening normality is another way to settle the matter.
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| ==References==
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| * [[Lynn Steen|Lynn Arthur Steen]] and J. Arthur Seebach, ''Counterexamples in Topology'', Dover Books, 1995. ISBN 0-486-68735-X
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| * {{PlanetMath attribution|id=6496|title=Moore space}}
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| * ''The original definition by [[R.L. Moore]] appears here'':
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| :: MR0150722 (27 #709) Moore, R. L. ''Foundations of point set theory''. Revised edition. American Mathematical Society Colloquium Publications, Vol. XIII American Mathematical Society, Providence, R.I. 1962 xi+419 pp. (Reviewer: F. Burton Jones)
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| * ''Historical information can be found here'':
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| :: MR0199840 (33 #7980) Jones, F. Burton "Metrization". ''[[American Mathematical Monthly]]'' 73 1966 571–576. (Reviewer: R. W. Bagley)
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| * ''Historical information can be found here'':
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| :: MR0203661 (34 #3510) Bing, R. H. "Challenging conjectures". ''American Mathematical Monthly'' 74 1967 no. 1, part II, 56–64;
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| * ''Vickery's theorem may be found here'':
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| :: MR0001909 (1,317f) Vickery, C. W. "Axioms for Moore spaces and metric spaces". ''Bulletin of the American Mathematical Society'' 46, (1940). 560–564
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| [[Category:General topology]]
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