Concerning Metrizability of Pointwise Paracompact Moore Spaces

1964 ◽  
Vol 16 ◽  
pp. 407-411 ◽  
Author(s):  
D. R. Traylor
Keyword(s):  
Moore Space ◽  
Moore Spaces

Although it is known that there exists a pointwise paracompact Moore space which is not metrizable (1), very little seems to be known about the metrizability of pointwise paracompact Moore spaces. This paper is devoted to determining some of the conditions under which a pointwise paracompact Moore space is metrizable.The statement that 5 is a Moore space means that there exists a sequence of collections of regions in 5 satisfying Axiom 0 and the first three parts of Axiom 1 of (2). A Moore space is complete if and only if it satisfies all of Axiom 1 of (2).

scholarly journals Metrisation of Moore spaces and abstract topological manifolds

1997 ◽  
Vol 56 (3) ◽  
pp. 395-401 ◽  
Author(s):  
David L. Fearnley
Keyword(s):  
Recent Work ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Moore Space ◽  
Global Coherence ◽  
Topological Manifolds ◽  
Necessary And Sufficient ◽  
Moore Spaces ◽  
Significant Class

The problem of metrising abstract topological spaces constitutes one of the major themes of topology. Since, for each new significant class of topological spaces this question arises, the problem is always current. One of the famous metrisation problems is the Normal Moore Space Conjecture. It is known from relatively recent work that one must add special conditions in order to be able to get affirmative results for this problem. In this paper we establish such special conditions. Since these conditions are characterised by local simplicity and global coherence they are referred to in this paper generically as “abstract topological manifolds.” In particular we establish a generalisation of a classical development of Bing, giving a proof which is complete in itself, not depending on the result or arguments of Bing. In addition we show that the spaces recently developed by Collins designated as “W satisfying open G(N)” are metrisable if they are locally separable and locally connected and regular. Finally, we establish a new necessary and sufficient condition for spaces to be metrisable.

Some Examples of Normal Moore Spaces

1977 ◽  
Vol 29 (1) ◽  
pp. 84-92 ◽  
Author(s):  
Mary Ellen Rudin ◽  
Michael Starbird
Keyword(s):  
Metric Space ◽  
Locally Compact ◽  
Moore Space ◽  
Uncountable Cardinal ◽  
Moore Spaces

A normal Moore space is non-metrizable only if it fails to be ƛ-collectionwise normal for some uncountable cardinal ƛ [1].For each uncountable cardinal X we present a class of normal, locally metrizable Moore spaces and a particular space Sλ in . If there is any space of class which is not X-collectionwise normal, then Sλ is such a space. The conditions for membership in make a space in behave like a subset of a product of a Moore space with a metric space. The class is sufficiently large to allow us to prove the following. Suppose F is a locally compact, 0-dimensional Moore space (not necessarily normal) with a basis of cardinality X and M is a metric space which is O-dimensional in the covering sense. If there is a normal, not X-collectionwise normal Moore space X where X ⊂ Y × M, then Sx is a normal, not λ-collectionwise normal Moore space.

On Screenability and Metrizability of Moore Spaces

1971 ◽  
Vol 23 (6) ◽  
pp. 1087-1092 ◽  
Author(s):  
G. M. Reed
Keyword(s):  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Point ◽  
Moore Space ◽  
Necessary And Sufficient ◽  
Moore Spaces

After showing that each screenable Moore space is pointwise paracompact and that the converse is not true, Heath in [4] asked for a necessary and sufficient condition for a pointwise paracompact Moore space to be screenable. In [12], Traylor asked for a necessary and sufficient condition for a pointwise paracompact Moore space to be metrizable. It is the purpose of this paper to provide such conditions, and to establish relationships between those conditions and metrization problems in Moore spaces.A Moore space S is a space (all spaces are T1) in which there exists a sequence G = (G1, G2, …) of open coverings of S, called a development, which satisfies the first three parts of Axiom I in [7]. The statement that a collection H of subsets of the space S is point finite (point countable) means that no point of S belongs to infinitely (uncountably) many elements of H.

Screenability, Pointwise Paracompactness, and Metrization of Moore Spaces

1964 ◽  
Vol 16 ◽  
pp. 763-770 ◽  
Author(s):  
R. W. Heath
Keyword(s):  
Moore Space ◽  
Moore Spaces ◽  
Made In

F. B. Jones (6) has shown that, if , then every separable normal Moore space is metrizable. It is not known whether this assumption is necessary, though perhaps some progress is made in (5). However, it is easily seen from R. H. Bing's Example E in (3) that a certain condition (see (1) below) implied by is necessary. Also in (3), Bing showed that every screenable normal Moore space is metrizable. In this paper we establish that: (1) every separable normal Moore space is metrizable if and only if every uncountable subspace M of E1 contains a subset which is not an Fσ (in M); (2) if every pointwise paracompact normal Moore space is metrizable, then so is every separable normal Moore space; (3) every screenable Moore space is pointwise paracompact but not conversely; (4) a T3-space is a pointwise paracompact Moore space if and only if it has a uniform base (in the sense of (1, p. 40), not a uniformity).

Co-H-Structures on Moore Spaces of Type (G, 2)

1994 ◽  
Vol 46 (4) ◽  
pp. 673-686 ◽  
Author(s):  
Martin Arkowitz ◽  
Marek Golasinski
Keyword(s):  
Abelian Group ◽  
Homotopy Classes ◽  
Moore Space ◽  
Moore Spaces

AbstractWe consider the set (of homotopy classes) of co-H-structures on a Moore space M(G,n), where G is an abelian group and n is an integer ≥ 2. It is shown that for n > 2 the set has one element and for n = 2 the set is in one-one correspondence with Ext(G, G ⊗ G). We make a detailed investigation of the co-H-structures on M(G, 2) in the case G = ℤm, the integers mod m. We give a specific indexing of the co-H-structures on M(ℤm, 2) and of the homotopy classes of maps from M(ℤm, 2) to M(ℤn, 2) by means of certain standard homotopy elements. In terms of this indexing we completely determine the co-H-maps from M{ℤm, 2) to M(ℤn, 2) for each co-H-structure on M(ℤm, 2) and on M(ℤn, 2). This enables us to describe the action of the group of homotopy equivalences of M(ℤn, 2) on the set of co-H-structures of M(ℤm, 2). We prove that the action is transitive. From this it follows that if m is odd, all co-H-structures on M(ℤm, 2) are associative and commutative, and if m is even, all co-H-structures on M(ℤm, 2) are associative and non-commutative.

scholarly journals ON DENSE EMBEDDINGS INTO MOORE SPACES WITH THE BAIRE PROPERTY

2010 ◽  
Vol 83 (1) ◽  
pp. 1-10
Author(s):  
DAVID L. FEARNLEY ◽  
LAWRENCE FEARNLEY
Keyword(s):  
Baire Property ◽  
Moore Space ◽  
Moore Spaces

AbstractWe demonstrate a construction that will densely embed a Moore space into a Moore space with the Baire property when this is possible. We also show how this technique generates a new ‘if and only if’ condition for determining when Moore spaces can be densely embedded in Moore spaces with the Baire property, and briefly discuss how this condition can can be used to generate new proofs that certain Moore spaces cannot be densely embedded in Moore spaces with the Baire property.

Moore Spaces, Semi-Metric Spaces and Continuous Mappings Connected with Them

1972 ◽  
Vol 24 (6) ◽  
pp. 1033-1042 ◽  
Author(s):  
C. M. Pareek
Keyword(s):  
Metric Spaces ◽  
Locally Compact ◽  
Moore Space ◽  
Paracompact Space ◽  
Continuous Mappings ◽  
Perfect Map ◽  
Moore Spaces

In [1] Arhangel'skiĭ announced that any σ-paracompact p-space could be mapped onto a Moore space by a perfect map. However Burke [3] recently showed that this is not true in general and he gave an example of a T2, locally compact, σ-paracompact space which cannot be mapped onto a Moore space by a perfect map.

Peripheral Covering Properties Imply Covering Properties

1968 ◽  
Vol 20 ◽  
pp. 257-263
Author(s):  
E. E. Grace
Keyword(s):  
Slight Variation ◽  
Covering Property ◽  
Moore Space ◽  
Covering Properties ◽  
Moore Spaces

Recently several papers (11; 12; 13; 14) have been published in which it is shown that a Moore space (normal, in one case) is metrizable if it has the peripheral version (in the sense defined below) of a certain covering property that was known to imply metrizability of Moore spaces. Each of these metrization theorems can be proved more easily by using a slight variation of the appropriate standard proof to show that such a space is collectionwise normal and hence (2, Theorem 10) metrizable. But this approach, as well as that followed in (11 ; 12; 13 ; 14), obscures the point that, in Moore spaces and in more general settings, the peripheral versions of these covering properties imply the covering properties.

James-Hopf Invariants, Anick’s Spaces, and the Double Loops on Odd Primary Moore Spaces

2000 ◽  
Vol 43 (2) ◽  
pp. 226-235
Author(s):  
Joseph Neisendorfer
Keyword(s):  
Homotopy Groups ◽  
Dimensional Sphere ◽  
Double Loop ◽  
Loop Spaces ◽  
Moore Space ◽  
Moore Spaces ◽  
Hopf Invariants

AbstractUsing spaces introduced by Anick, we construct a decomposition into indecomposable factors of the double loop spaces of odd primary Moore spaces when the powers of the primes are greater than the first power. If n is greater than 1, this implies that the odd primary part of all the homotopy groups of the 2n + 1 dimensional sphere lifts to a mod pr Moore space.

Sign in / Sign up

Export Citation Format

Share Document