In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, more specifically point-set topology, a Moore space is a developable regular Hausdorff space. Equivalently, a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 X is a Moore space if the following conditions hold:

• Any two distinct points can be separated by neighbourhoods, and any closed set
  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...
  
   and any point in its complement
  Complement (set theory)
  In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
  
   can be separated by neighbourhoods. (X is a regular Hausdorff space.)
• There is a countable
  Countable set
  In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
  
   collection of open covers 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

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 is said to be metrizable if there is a metricd\colon X \times X \to [0,\infty)...

s. The concept of a Moore space was formulated by R. L. Moore in the earlier part of the 20th century.

Examples and properties

1. Every metrizable space, X, is a Moore space. If {A_x} is an open cover of X (indexed by 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.
2. Moore spaces are a lot like regular spaces and different to normal space
   Normal space
   In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...
   
   s in the sense that every subspace of a Moore space is also a Moore space.
3. 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.
4. Both examples 2 and 3 suggest that Moore spaces are a lot similar to regular spaces.
5. Neither the Sorgenfrey line nor the Sorgenfrey plane are Moore spaces because they are normal and not second countable.
6. The Moore plane
   Moore plane
   In mathematics, the Moore plane, also sometimes called Niemytzki plane is a topological space. It is a completely regular Hausdorff space which is not normal...
   
    (also known as the Niemytski space) is an example of a non-metrizable Moore space.
7. Every metacompact, separable, normal Moore space is metrizable. This theorem is known as Traylor’s theorem.
8. Every locally compact
   Locally compact space
   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.-Formal definition:...
   
   , locally connected space, normal Moore space is metrizable. This theorem was proved by Reed and Zenor.
9. If , then every separable normal
   Normal space
   In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...
   
    Moore space is metrizable. This theorem is known as Jones’ theorem.

History

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. There were some nice partial results at first; namely properties 7, 8 and 9 as given in the previous section.

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 axiom of constructibility implies that locally compact, normal Moore spaces are metrizable.

On the other hand, under the Continuum hypothesis

Continuum hypothesis

In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...

 (CH) and also under Martin's Axiom

Martin's axiom

In the mathematical field of set theory, Martin's axiom, introduced by , is a statement which is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, so certainly consistent with ZFC, but is also known to be consistent with ZF + ¬ CH...

 and not CH, there are several examples of non-metrizable normal Moore spaces. Finally, Nyikos settled the matter: the conjecture is independent

Independence (mathematical logic)

In mathematical logic, independence refers to the unprovability of a sentence from other sentences.A sentence σ is independent of a given first-order theory T if T neither proves nor refutes σ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T that...

 of ZFC. Under the so-called PMEA (Product Measure Extension Axiom), which needs a large cardinal, all normal Moore spaces are metrizable. Moreover, 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.

Moore

Robert Lee Moore

Robert Lee Moore was an American mathematician, known for his work in general topology and the Moore method of teaching university mathematics.-Life:...

 himself proved the theorem that a collectionwise normal

Collectionwise normal

In mathematics, a topological space X is called collectionwise normal if for every discrete family Fi of closed subsets of X there exists a pairwise disjoint family of open sets Ui , such that Fi ⊂ Ui...

Moore space is metrizable, so strengthening normality is another way to settle the matter.