HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, more specifically
point-set topology In mathematics, general topology (or point set 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 differ ...
, a Moore space is a developable regular Hausdorff space. That is, a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
''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 (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...
and any point in its complement can be separated by neighbourhoods. (''X'' is a regular Hausdorff space.) * There is a
countable In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
collection of
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a family 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\su ...
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 theorems. The concept of a Moore space was formulated by R. L. Moore in the earlier part of the 20th century.


Examples and properties

#Every
metrizable space In topology and related areas of mathematics, a metrizable space is a topological space that is Homeomorphism, homeomorphic to a metric space. That is, a topological space (X, \tau) is said to be metrizable if there is a Metric (mathematics), metr ...
, ''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 space Normal(s) or The Normal(s) may refer to: Film and television * Normal (2003 film), ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * Normal (2007 film), ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keit ...
s 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. #The Moore plane (also known as the Niemytski plane) 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 e ...
,
locally connected In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting of open connected sets. As a stronger notion, the space ''X'' is locally path connected if ev ...
normal Moore space is metrizable. This theorem was proved by Reed and Zenor. #If 2^<2^, then every separable normal Moore space is
metrizable 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, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...
. 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 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, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...
. 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. 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 In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: Or equivalently: In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this ...
(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 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 ...
, 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 himself proved the theorem that a collectionwise normal Moore space is metrizable, so strengthening normality is another way to settle the matter.


References

* Lynn Arthur Steen and J. Arthur Seebach, ''Counterexamples in Topology'', Dover Books, 1995. *. * . * ''The original definition by R.L. Moore 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 peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposi ...
'' 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