Klaas Pieter Hart

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

Discrete sets and the maximal totally bounded group topology

Hart, K.P. and J. van Mill, Discrete sets and the maximal totally bounded group topology, Journal of Pure and Applied Algebra 70 (1991) 73-80. If G is an Abelian group, then G # is G with its maximal totally bounded group topology. We prove that every A c G# contains a closed (in G#) and discrete subset B such that lB1 = IAl. This answers a question posed by Eric van Douwen. We also present an example of a countable G’ having an infinite relatively discrete subset that is not closed.

A note on monotonically metacompact spaces

Abstract We 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 -closed-discrete 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.

Applications of another characterization of βN⧹N

Steprprovided a characterization of βN\N in the ℵ2-Cohen model that is much in the spirit of Paroviy cenkos 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)/finthat feature in Steprresult.  2002 Elsevier Science B.V. All rights reserved. AMS classification:Primary 54A35, Secondary 03E35; 06E05; 54D35; 54F65

A note on an example by van Mill

Abstract Adapting an earlier example by J. van Mill, we prove that there exists a zero-dimensional compact space of countable π-weight and uncountable character that is homogeneous under MA + ¬ CH , but not under CH .

Crowded rational ultrafilters

Abstract We prove that if every family in ( ω ω,≤ ∗ ) of size less than c is bounded then there exists a point p in Q ∗ such that p generates an ultrafilter in the set-theoretic sense on Q and such that p has a base consisting of sets that are homeomorphic to Q . This is a partial answer to Question 30 (Problem 229) in (Hart and van Mill, 1990).

Span, chainability and the continua H ∗ and Iu

Abstract We show that the continua I u and H ∗ are nonchainable and have span nonzero. Under CH this can be strengthened to surjective symmetric span nonzero. We discuss the logical consequences of this.

Admissibility, homeomorphism extension and the AR-property in topological linear spaces

Abstract We consider topological linear spaces (without local convexity) and their convex subsets. We investigate relations between admissibility, the AR-property and the possibility of extending homeomorphisms between compacta.

Elementary chains and compact spaces with a small diagonal

Abstract It is a well known open problem if, in ZFC , each compact space with a small diagonal is metrizable. We explore properties of compact spaces with a small diagonal using elementary chains of substructures. We prove that ccc subspaces of such spaces have countable π -weight. We generalize a result of Gruenhage about spaces which are metrizably fibered. Finally we discover that if there is a Luzin set of reals, then every compact space with a small diagonal will have many points of countable character.

A small transitive family of continuous functions on the Cantor set

Abstract In this paper we show that, when we iteratively add Sacks reals to a model of ZFC we have for every two reals in the extension a continuous function defined in the ground model that maps one of the reals to the other.