Steprāns provided a characterization of βN \ N in the ℵ 2 -Cohen model that is much in the spirit of Parovičenko's characterization of this space under CH. A variety of the topological results established in the Cohen model can be deduced directly from the properties of βN \ N or P(N)/fin that feature in Steprāns' result.Cohen reals. 'The Cohen model' is any model obtained from a model of the GCH by adding a substantial quantity of Cohen reals -more than ℵ 1 . In particular 'the ℵ 2 -Cohen model' is obtained by adding ℵ 2 many Cohen reals. Actually, since we are intent on proving our results using the properties of P(N)/fin only, many readers may elect to take Lemma 2.2, Theorem 2.7 and the remark made after Proposition 2.12 on faith or else consult [18] for the necessary background on Cohen forcing.The weak Freese-Nation property. A partially ordered set P is said to have the weak Freese-Nation property if there is a function F : P → [P ] ℵ0 such that whenever p q there is r ∈ F (p) ∩ F (q) with p r q.Elementary substructures. Consider two structures M and N (groups, fields, Boolean algebras, models of set theory . . . ), where M is a substructure of N . We say that M is an elementary substructure of N , and we write M ≺ N , if every equation, involving the relations and operations of the structures and constants from M , that has a solution in N has a solution in M as well.

Using MAcountable we construct a topological group with the properties mentioned in the title. 0. Introduction In this paper we construct, assuming Martin's Axiom for countable posets (MAcountable), a countably compact topological group 77 for which 77 is not countably compact. The existence of such a group under MA was announced by van Douwen in [vD] but was never published. Our construction is like van Douwen's construction in [vD] in that our group is a subgroup of c2 and that we construct the points of our group in an induction of length c. But whereas van Douwen used MA to construct a countably compact subgroup of c2 without convergent sequences and then constructed two countably compact subgroups of this group with a countable intersection, we construct our group all at once. Moreover, our group 77 can be written as G + D with D a countable subgroup and G an eobounded subgroup, so that our group has many convergent sequences. Also van Douwen needed MA for certain uncountable posets, we get by using MAcountable only. The interest in examples such as the ones in the present paper comes from the fact that by Comfort and Ross [CR] the product of an arbitrary family of pseudocompact topological groups is pseudocompact. (For more information see [vD].) This paper is organized as follows: §1 contains some definitions and preliminaries. In §2 we use our technique to construct two countably compact subgroups 770 and 77j of c2 such that 770 x Hx is not countably compact, and in §3 we construct our main example 77. The reason for doing this is that the construction in §3 is rather involved notationwise so that it may be useful to have a more transparant version available. Also, in §3, we essentially construct a countably compact subgroup 77 of £2x'2 which contains (modified

Abstract. We prove that every continuum of weight ℵ 1 is a continuous image of theČech-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 ω 2 + 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 ultrafilter.We also show that a universal continuum can be gotten from a c-saturated ultrafilter on ω, and that it is consistent that there is no universal continuum of weight c.

MSC: primary 54D20 secondary 54E30, 54E35, 54F05 Keywords: Metacompact Countably metacompact Monotonically metacompact Monotonically countably metacompact Generalized ordered space GO-space LOTS Metacompact Moore space Metrizable σ -Closed-discrete dense setWe show that any metacompact Moore space is monotonically metacompact and use that result to characterize monotone metacompactness in certain generalized ordered (GO) spaces. We show, for example, that a generalized ordered space with a σ -closeddiscrete dense subset is metrizable if and only if it is monotonically (countably) metacompact, that a monotonically (countably) metacompact GO-space is hereditarily paracompact, and that a locally countably compact GO-space is metrizable if and only if it is monotonically (countably) metacompact. We give an example of a non-metrizable LOTS that is monotonically metacompact, thereby answering a question posed by S.G. Popvassilev. We also give consistent examples showing that if there is a Souslin line, then there is one Souslin line that is monotonically countable metacompact, and another Souslin line that is not monotonically countably metacompact.

We prove that the following statement follows from the Open Colouring Axiom (OCA): if X is locally compact σ-compact but not compact and if itsČech-Stone remainder X * is a continuous image of ω * then X is the union of ω 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 of ω * .

The Open Colouring Axiom implies that the measure algebra cannot be embedded into P(N)/fin. We also discuss errors in previous results on the embeddability of the measure algebra. Introduction. The aim of this paper is to prove the following result. Main Theorem. The Open Colouring Axiom implies that the measure algebra cannot be embedded into the Boolean algebra P(N)/fin. By "the measure algebra" we mean the quotient of the σ-algebra of Borel sets of the real line by the ideal of sets of measure zero. There are various reasons, besides sheer curiosity, why it is of interest to know whether the measure algebra can be embedded into P(N)/fin. One reason is that there is great interest in determining what the subalgebras of P(N)/fin are. One of the earliest and most influential results in this direction is Parovichenko's theorem from [14], which states that every Boolean algebra of size ℵ 1 can be embedded into P(N)/fin, with the obvious corollary that the Continuum Hypothesis (CH) implies that P(N)/fin is a universal Boolean algebra of size c: a Boolean algebra embeds into P(N)/fin iff it is of size c or less. It is therefore natural to ask how much of this universality remains without assumptions beyond ZFC. It has long been known that every σ-centered Boolean algebra embeds into P(N)/fin but the question for more general c.c.c. Boolean algebras has proven to be much more difficult-with the case of the measure algebra being seen as a touchstone.

Using side-by-side Sacks forcing, it is shown that it is consistent that 2ω be large and that there be many types of ultrafilters of character ω1.

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