Normal Moore Space Conjecture Research Articles

Weak separation axioms and weak covering properties

We identify some remnants of normality and call them rudimentary normality, generalize the concept of submetacompact spaces to that of a weakly subparacompact space and that of a weakly ⁎ subparacompact space, and make a simultaneous generalization of collectionwise normality and screenability with the introduction of what is to be called collectionwise σ-normality. With these weak properties, we show that, 1) on weakly subparacompact spaces, countable compactness = compactness, ω 1 -compactness = Lindelöfness; 2) on weakly subparacompact Hausdorff spaces with rudimentary normality, regularity = normality = countable paracompactness; and 3) on weakly subparacompact regular T 1 -spaces with rudimentary normality, collectionwise σ-normality = screenability = collectionwise normality = paracompactness. The famous Normal Moore Space Conjecture is thus given an even more striking appearance and Worrell and Wickeʼs factorization of paracompactness (over Hausdorff spaces) along with Krajewskiʼs are combined and strengthened. The methodology extends itself to the factorization of paracompactness on locally compact, locally connected spaces in the manner of Gruenhage and on locally compact spaces in that of Tall, and to the factorization of subparacompactness and metacompactness in the genre of Katuta, Chaber, Junnila and Price and Smith and that of Boone, improving all of them.

Peter J. Nyikos. A provisional solution to the normal Moore space problem. Proceedings of the American Mathematical Society, vol. 78 (1980), pp. 429–435. - William G. Fleissner. If all normal Moore spaces are metrizable, then there is an inner model with a measurable cardinal. Transactions of the American Mathematical Society, vol. 273 (1982), pp. 365–373. - Alan Dow, Franklin D.

Peter J. Nyikos. A provisional solution to the normal Moore space problem. Proceedings of the American Mathematical Society, vol. 78 (1980), pp. 429–435. - William G. Fleissner. If all normal Moore spaces are metrizable, then there is an inner model with a measurable cardinal. Transactions of the American Mathematical Society, vol. 273 (1982), pp. 365–373. - Alan Dow, Franklin D. Tall, and William A. R. Weiss. New proofs of the consistency of the normal Moore space conjecture I. Topology and its applications, vol. 37 (1990), pp. 33–51. - Zoltán Balogh. On collectionwise normality of locally compact, normal spaces. Transactions of the American Mathematical Society, vol. 323 (1991), pp. 389–411. - Volume 8 Issue 3

Metrisation of Moore spaces and abstract topological manifolds

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.

New proofs of the consistency of the normal Moore space conjecture II

This article continues our first paper on this topic. In this article we first continue our review of the technique of iterated forcing and reflection by describing the machinery developed for reflection from a weakly compact cardinal. Next we present more applications of the technique to show conditions under which nonmetrizability, nonparacompactness, or nondevelopability reflect. In the final section we present yet another proof of the normal Moore space conjecture which avoids elementary embeddings by using filter combinatorics and provides a quick path to the solution for those less interested in generally applicable techniques.

New proofs of the consistency of the normal moore space conjecture I

The normal Moore space conjecture asserts that normal Moore spaces are metrizable. Nyikos has proven the consistency (from the existence of a strongly compact cardinal) of the conjecture holding and Fleissner has proven that at least a measurable cardinal is needed to prove the consistency. Although extremely elegant, Nyikos' proof relies on Kunen's proof of the consistency of the product measure extension axiom and does not lend itself to other applications. In this paper we first present the groundwork for iterated forcing and reflection type proofs from the assumption of a supercompact cardinal. We then use this technology to give a proof of the normal Moore space conjecture as well as several other similar results which use a variation of the proof.

PMEA implies proposition P

The Product Measure Extension Axiom (PMEA) asserts that for every set A, Haar measure on 2 A can be extended to all subsets of 2 A . PMEA implies the normal Moore space conjecture. Proposition P is the statement that every point-finite analytic-additive family of subsets of a metrizable space is σ-discretely decomposible. Proposition P is useful in nonseparable Borel theory. We show in this paper that PMEA implies Proposition P.

On the metrizability of $F_pp$-spaces and its relationship to the normal Moore space conjecture

On the metrizability of $F_pp$-spaces and its relationship to the normal Moore space conjecture

A Note on the Normal Moore Space Conjecture

F. B. Jones (1937) conjectured that every normal Moore space is metrizable. He also denned a particular kind of topological space (now known as Jones' spaces), proved that they were all non-metrizable Moore spaces, but was unable to decide whether or not Jones’ spaces are normal. J. H. Silver (1967) proved that a positive solution to Jones’ conjecture was not possible, and W. Fleissner (1973) obtained an alternative proof by showing that it is not possible to prove the non-normality of Jones’ spaces. These results left open the possibility of resolving the questions from the GCH. In this paper we show that if CH be assumed, then Jones’ spaces are not normal (Devlin, Shelah, independently) and that the GCH does not lead to a positive solution to the Jones conjecture (Shelah). A brief survey of the progress on the problem to date is also included.

Countable paracompactness, normality, and Moore spaces

In this paper we show that MA + ¬ CH {\text {MA}} + \neg {\text {CH}} implies that there exists a countably paracompact Moore space which is not normal. Further, if there is a model of set theory in which every countably paracompact Moore space is normal, then the normal Moore space conjecture is true in that model. Other examples are given, including a nonnormal space constructed with ⋄ \diamond which is countably compact, T 3 {T_3} , first countable, locally compact, perfect, and hereditarily separable.

A topological translation of a fundamental problem in cluster set theory

Two conjectures related to the problem of finding a “large” family of approach curves along which cluster sets can be preassigned are shown to be equivalent to special cases of the normal Moore space conjecture.

Normal Moore Spaces in the Constructible Universe

Assuming the axiom of constructibility, points in closed discrete subspaces of certain normal spaces can be simultaneously separated. This is a partial result towards the normal Moore space conjecture.

Normal Moore spaces in the constructible universe

Assuming the axiom of constructibility, points in closed discrete subspaces of certain normal spaces can be simultaneously separated. This is a partial result towards the normal Moore space conjecture.

Concerning normality, metrizability and the Souslin property in subspaces of Moore spaces

In this paper, the author generalizes the known results concerning the existence of dense metrizable subspaces in normal Moore spaces and characterizes those Moore spaces which have certain types of dense metrizable subspaces. An example is given of a Moore space which has a dense metrizable subspace but for which there exists no development satisfying Axiom C at each point of a dense subset. The concepts developed are also used to investigate various problems related to the normal Moore space conjecture of F.B. Jones and to characterize the Souslin property in Moore spaces.

A Translation of the Normal Moore Space Conjecture

A Translation of the Normal Moore Space Conjecture

A translation of the normal Moore space conjecture

A translation of the normal Moore space conjecture