Normal Moore Space Research Articles

ω1-strongly compact cardinals and normality

We present more applications of the recently introduced ω1-strongly compact cardinals in the context of either consistency or reflection results in General Topology, focusing on issues related to normality. In particular, we show that such large cardinal notion provides a new upper bound for the consistency strength of the statement “All normal Moore spaces are metrizable” (NMSC). The proof uses random forcing, as in the original consistency proof of NMSC due to Nykos-Kunen-Solovay (see Fleissner [10]). We establish a compactness theorem for normality (i.e., reflection of non-normality) in the realm of first countable spaces, using the least ω1-strongly compact cardinal, as well as two more similar compactness results on related topological properties. We finish the paper by combining the techniques of reflection and forcing to show that our new upper bound for the consistency strength of NMSC can be also obtained via Cohen forcing, using some arguments from Dow-Tall-Weiss [6].

Metrizability of partial metric spaces

We analyze relationship between partial metric spaces and several generalized metric spaces. We first establish that the perfectness of an arbitrary partially metric space X is equivalent to each of the following properties: developability, semi-stratifiability, X has a σ-discrete closed network, X is a β-space with T1. Using this fact we obtain that the metrizability of a partial metric space X is equivalent to the stratifiability of X or to the fact that X is a perfect regular paracompact space. Moreover, we give examples that indicate the essentiality of certain conditions in the previous two results. Finally, we show that the statement “every perfectly normal separable partial metric space is metrizable” is independent of ZFC, similarly as for the Normal Moore Space Problem.

A study of chain conditions and dually properties

In this paper, we make some observations on chain conditions and dually properties. In particular, we show that:(1) A subspace X⊂ω1ω is dually CCC then e(X)≤ω and a normal subspace X⊂ω1ω is DCCC if and only if e(X)≤ω;(2) There is a Tychonoff pseudocompact subspace X⊂(ω1+1)2 which is not dually CCC;(3) In the class of o-semimetrizable spaces, dually separable is self-dual with respect to neighbourhood assignments. As an application, we obtain an example of a CCC normal Moore space which is not dually separable under MA+¬CH;(4) There exists an example of a large normal CCC semi-stratifiable space, which answers a question of Xuan and Song (2018) [21, Question 4.11];(5) Every dually separable and monotonically monolithic space is Lindelöf, which gives a partial answer to a question of Alas, Junqueira, van Mill, Tkachuk and Wilson (2011) [2, Question 2.1];(6) A dually separable Hausdorff space with a strong rank 1-diagonal has cardinality at most 2c. The conclusion is also true for regular spaces if we replace “strong rank 1-diagonal” with “Gδ-diagonal”;(7) A dually separable ω-monolithic Hausdorff space with a Gδ-diagonal has cardinality at most c.

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

A Provisional Solution to the Normal Moore Space Problem

A Provisional Solution to the Normal Moore Space Problem

On the non-productivity of normality in Moore spaces

Under Martin’s Axiom and the denial of the Continuum Hypothesis, the authors give examples of normal Moore spaces whose squares are not normal.

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.

𝜎-centred forcing and reflection of (sub)metrizability

By using supercompact reflection and preservation lemmas for random real forcing and σ \sigma -centred forcing, we obtain a model in which every normal Moore space is submetrizable, but not every normal Moore space is metrizable.

Topological applications of generic huge embeddings

In the Foreman-Laver model obtained by huge cardinal collapse, for many $\Phi ,\Phi ({\aleph _1})$ implies $\Phi ({\aleph _2})$. There are a variety of set-theoretic and topological applications, in particular to paracompactness. The key tools are generic huge embeddings and preservation via $\kappa$-centred forcing. We also formulate "potent axioms" à la Foreman which enable us to transfer from ${\aleph _1}$ to all cardinals. One such axiom implies that all ${\aleph _1}$-collectionwise normal Moore spaces are metrizable. It also implies (as does Martin’s Maximum) that a first countable generalized ordered space is hereditarily paracompact iff every subspace of size ${\aleph _1}$ is paracompact.

Two Moore manifolds

Reed and Zenor have shown that locally connected, locally compact, normal Moore spaces are metrizable. The first of the two examples presented is a locally connected, locally compact, pseudonormal nonmetrizable Moore space. The second is a locally connected, locally compact, pseudocompact nonmetrizable Moore space and can be constructed assuming the Continuum Hypothesis. Therefore normality in the Reed-Zenor theorem cannot be replaced by pseudonormality or (consistently) pseudocompactness. Both spaces can be modified in such a way that they are manifolds.

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.

Covering and separation properties in the Easton model

The “normality implies collectionwise normality” results known in the reverse Easton model are shown to hold when normality is replaced by countable paracompactness plus submetacompactness. Moreover, the reverse Easton model can be replaced by the Easton model. A sidelight of the proof is the consistency with GCH of “every normal Moore space of cardinality ⩽ℵ 1 is paralindelöf”.

Paracompactness in Locally Lindelöf Spaces

This paper contains a set of results concerning paracompactness of locally nice spaces which can be proved by (variations on) the technique of “stationary sets and chaining” combined with other techniques available at the present stage of knowledge in the field. The material covered by the paper is arranged in three sections, each containing, in essence, one main result.The main result of Section 1 says that a locally Lindelöf, submeta-Lindelöf ( = δθ-refinable) space is paracompact if and only if it is strongly collectionwise Hausdorff. Two consequences of this theorem, respectively, answer a question raised by Tall [7], and strengthen a result of Watson [9]. In the last two sections, connected spaces are dealt with. The main result of the second section can be best understood from one of its consequences which says that under 2ωl > 2ω, connected, locally Lindelöf, normal Moore spaces are metrizable.

If all Normal Moore Spaces are Metrizable, then there is an Inner Model with a Measurable Cardinal

If all Normal Moore Spaces are Metrizable, then there is an Inner Model with a Measurable Cardinal

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.

Normal nonmetrizable Moore space from continuum hypothesis or nonexistence of inner models with measurable cardinals

Assuming the continuum hypothesis, a normal nonmetrizable Moore space is constructed. This answers a question raised by F. B. Jones in 1931, using an axiom well known at that time. For the construction, a consequence of the continuum hypothesis that also follows from the nonexistence of an inner model with a measurable cardinal is used. Hence, it is shown that to prove the consistency of the statement that all normal Moore spaces are metrizable one must assume the consistency of the statement that measurable cardinals exist.

If all normal Moore spaces are metrizable, then there is an inner model with a measurable cardinal

We formulate an axiom, HYP, and from it construct a normal, metacompact, nonmetrizable Moore space. HYP unifies two better known axioms. The Continuum Hypothesis implies HYP; the nonexistence of an inner model with a measurable cardinal implies HYP. As a consequence, it is impossible to replace Nyikos’ "provisional" solution to the normal Moore space problem with a solution not involving large cardinals. Finally, we discuss how this space continues a process of lowering the character for normal, not collectionwise normal spaces.

Diamond Principles, Ideals and the Normal Moore Space Problem

If is a topological space then a sequence (Cα:α < λ) of subsets of is said to be normalized if for every H ⊆ λ there exist disjoint open sets and such thatThe sequence (Cα:α < λ) is said to be separated if there exists a sequence of pairwise disjoint open sets such that for each α < λ. As is customary, we adopt the convention that all sequences (Cα:α < λ) considered are assumed to be relatively discrete as defined in [18, p. 21]: if x ∈ Cα then there exists a neighborhood about x that intersects no Cβ for β ≠ α.