Alan Dow

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

Abstract 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.

Two classes of Fréchet-Urysohn spaces

Arhangelskii introduced five classes of spaces, a, -spaces (i < 5), which are important in the study of products of Frechet-Urysohn spaces. For each i < 5, each a, -space is an ai+ I-space and it follows from the continuum hypothesis that there are countable a,+,-spaces which are not a,-spaces. A v-space ( w-space) is a Frechet-Urysohn a, -space ( a2-space). We show that there is a model of set theory in which each a2-space ( w-space) is an a,-space (v-space).

Homogeneity in powers of zero-dimensional first-countable spaces

A construction of L. Brian Lawrence is extended to show that the w-power of every subset of the Cantor set is homogeneous via a continuous translation modulo a dense set. It follows that every zero-dimensional firstcountable space has a homogeneous w-power.

Remote points in large products

Abstract A point p ∈ β X \ X is a remote point of X if p ∉ cl β X D for any nowhere dense D ⊂ X . Van Douwen, and independently Chae and Smith, have shown that each non-pseudocompact space of countable π-weight has a remote point. Van Mill showed that many spaces of π-weight ω 1 , such as ω×2 ω 1 also have remote points. We show that arbitrarily large products of spaces with countable π-weight which are not pseudocompact have remote points. In particular, ω×2 ϰ for any infinite cardinal ϰ.

More set-theory for topologists

Abstract Basic applications in topology of ◊, elementary submodels and forcing are illustrated. Finite and countable supported forcing iterations are reviewed. Also there are two new applications of Prikry forcing to preservation results.

A Universal Continuum of Weight aleph

We prove that every continuum of weight א1 is a continuous image of the Cech-Stone-remainder R∗ of the real line. It follows that under CH the remainder of the half line [0,∞) is universal among the continua of weight c — universal in the ‘mapping onto’ sense. We complement this result by showing that 1) under MA every continuum of weight less than c is a continuous image of R∗, 2) in the Cohen model the long segment of length ω2 + 1 is not a continuous image of R∗, and 3) PFA implies that Iu is not a continuous image of R∗, whenever u is a c-saturated

An empty class of nonmetric spaces

Let CSSM be the class of compact nonmetrizable spaces in which every subspace of cardinality at most oji is metrizable. We show that CSSM is empty. For the purposes of this article only let us call a space an SSM space (small subspaces metrizable) if it is not metrizable but it is regular and all of its subspaces of cardinality at most uii are metrizable. A CSSM space is a compact SSM space. If X is a CSSM space, then X is first countable (HJ). Therefore under the continuum hypothesis (CH) there are no CSSM spaces because, of course, a compact first countable space has cardinality at most c. This was first observed by Juhasz who then asked if the CH assumption could be removed (J). It was shown in (D) that it is consistent with (and independent of) ~1 CH that there are no Lindelof, countably compact or even wi-compact first countable SSM spaces. In this article we show that there simply are never any CSSM spaces. There is however an easy example, under MA+HCH, of a Lindelof first countable SSM space. I do not know if a Lindelof SSM space is necessarily first countable. EXAMPLE 1. Recall that the Alexandroff double topology on 7 x 2 (where 7 is the unit interval) is obtained by declaring 7 x {1} to be open and discrete while a basic open neighbourhood of a point (r, 0) is U x 2 — {(r, 1)} where r G U is open in 7. If A C 7 is any uncountable set containing no uncountable closed set, then X = (7 — A x {0}) U (A x {1}) is a Lindelof non metrizable subspace of the Alexandroff double. Furthermore, if MA(cji) is assumed then X is an SSM space since A x {1} will be an Fq—set in any subspace of X of cardinality uii (see (M)). One might hope to modify the Alexandroff double somehow to obtain a CSSM space. In fact if X were a CSSM space then X would contain an uncountable discrete subset D; hence cl D would itself be a CSSM space. (To see that X would contain such a D see 2(h).)

Thin-tall Boolean algebras and their automorphism groups

Our main result is a general construction to show that for each countable groupG there are 2ω1 pairwise non-isomorphic thin-tall superatomic Boolean algebras, all of them representingG as a group of non-trivial automorphisms.

ω* has (almost) no continuous images

We prove that the following statement follows from the Open Colouring Axiom (OCA): ifX is locally compactσ-compact but not compact and if its Čech-Stone remainderX* is a continuous image ofω*, thenX is the union ofω and a compact set. It follows that the remainders of familiar spaces like the real line or the sum of countably many Cantor sets need not be continuous images ofω*.

On the consistency of the Moore- Mrówka solution

Abstract We show that a large cardinal is not necessary to prove the consistency of the recent PFA results of Balogh, Fremlin and Nyikos concerning countably compact spaces of countable tightness and closed preimages of ω 1 . For example, we improve Baloghs result by showing that it is consistent that all compact spaces of countable tightness are sequential.