Moore space (topology)
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, more specifically
point-set topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
, a Moore space is a developable
regular Hausdorff space In topology and related fields of mathematics, a topological space ''X'' is called a regular space if every closed subset ''C'' of ''X'' and a point ''p'' not contained in ''C'' admit non-overlapping open neighborhoods. Thus ''p'' and ''C'' can ...
. That is, a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
''X'' is a Moore space if the following conditions hold: * Any two distinct points can be
separated by neighbourhoods In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets a ...
, and any
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
and any point in its complement can be separated by neighbourhoods. (''X'' is a
regular Hausdorff space In topology and related fields of mathematics, a topological space ''X'' is called a regular space if every closed subset ''C'' of ''X'' and a point ''p'' not contained in ''C'' admit non-overlapping open neighborhoods. Thus ''p'' and ''C'' can ...
.) * There is a countable collection of
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ...
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 from ''C''. (''X'' is a developable space.) Moore spaces are generally interesting in mathematics because they may be applied to prove interesting
metrization theorem In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
s. The concept of a Moore space was formulated by
R. L. Moore Robert Lee Moore (November 14, 1882 – October 4, 1974) was an American mathematician who taught for many years at the University of Texas. He is known for his work in general topology, for the Moore method of teaching university mathematics, ...
in the earlier part of the 20th century.


Examples and properties

#Every metrizable space, ''X'', is a Moore space. If 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. #Moore spaces are a lot like regular spaces and different from normal spaces in the sense that every subspace of a Moore space is also a Moore space. #The image of a Moore space under an injective, continuous open map is always a Moore space. (The image of a regular space under an injective, continuous open map is always regular.) #Both examples 2 and 3 suggest that Moore spaces are similar to regular spaces. #Neither the Sorgenfrey line nor the Sorgenfrey plane are Moore spaces because they are normal and not
second countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
. #The Moore plane (also known as the Niemytski space) is an example of a non-metrizable Moore space. #Every metacompact, separable, normal Moore space is metrizable. This theorem is known as Traylor’s theorem. #Every
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
, locally connected normal Moore space is metrizable. This theorem was proved by Reed and Zenor. #If 2^<2^, then every separable normal Moore space is metrizable. This theorem is known as Jones’ theorem.


Normal Moore space conjecture

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 theorem In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
. There were some nice partial results at first; namely properties 7, 8 and 9 as given in the previous section. With property 9, we see that we can 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 axiom of constructibility implies that locally compact, normal Moore spaces are metrizable. On the other hand, under the continuum hypothesis (CH) and also under
Martin's axiom In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consist ...
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 In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least Î ...
, 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. gave an example of a pseudonormal Moore space that is not metrizable, so the conjecture cannot be strengthened in this way.
Moore Moore may refer to: People * Moore (surname) ** List of people with surname Moore * Moore Crosthwaite (1907–1989), a British diplomat and ambassador * Moore Disney (1765–1846), a senior officer in the British Army * Moore Powell (died c. 1573 ...
himself proved the theorem that a
collectionwise normal In mathematics, a topological space X is called collectionwise normal if for every discrete family ''F'i'' (''i'' ∈ ''I'') of closed subsets of X there exists a pairwise disjoint family of open sets ''U'i'' (''i'' ∈ ''I''), such th ...
Moore space is metrizable, so strengthening normality is another way to settle the matter.


References

*
Lynn Arthur Steen Lynn Arthur Steen (January 1, 1941 – June 21, 2015) was an American mathematician who was a Professor of Mathematics at St. Olaf College, Northfield, Minnesota in the U.S. He wrote numerous books and articles on the teaching of mathematics. H ...
and J. Arthur Seebach, ''Counterexamples in Topology'', Dover Books, 1995. *. * . * ''The original definition by
R.L. Moore Robert Lee Moore (November 14, 1882 – October 4, 1974) was an American mathematician who taught for many years at the University of Texas. He is known for his work in general topology, for the Moore method of teaching university mathematics, ...
appears here'': :: (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) * ''Historical information can be found here'': :: (33 #7980) Jones, F. Burton "Metrization". ''
American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an e ...
'' 73 1966 571–576. (Reviewer: R. W. Bagley) * ''Historical information can be found here'': :: (34 #3510) Bing, R. H. "Challenging conjectures". ''American Mathematical Monthly'' 74 1967 no. 1, part II, 56–64; * ''Vickery's theorem may be found here'': :: (1,317f) Vickery, C. W. "Axioms for Moore spaces and metric spaces". ''Bulletin of the American Mathematical Society'' 46, (1940). 560–564 * {{PlanetMath attribution, id=6496, title=Moore space General topology Independence results