We give an alternative proof of Fedorchuk’s recent result that dimX6DgX for compact Hausdorff spaces X. We use the Löwenheim-Skolem theorem to reduce the problem to the metric case.

We prove that under reasonable assumptions, every cat (compact abstract theory) is metric, and develop some of the theory of metric cats. We generalise Morley’s theorem: if a countable Hausdorff cat T has a unique complete model of density character λ ≥ ω1, then it has a unique complete model of density character λ for every λ ≥ ω1.

The purpose of this paper is to prove directly that if two locally compact Hausdorff étale groupoids are Morita equivalent, then their reduced groupoid C∗-algebras are Morita equivalent.

In this paper we introduce and study so-called k∗-metrizable spaces forming a new class of generalized metric spaces, and display various applications of such spaces in topological algebra, functional analysis, and measure theory. By definition, a Hausdorff topological space X is k∗-metrizable if X is the image of a metrizable space M under a continuous map f : M → X having a section s : X → M ...

In the moduli space of quadratic differentials over complex structures on a surface, we construct a set of full Hausdorff dimension of points with bounded Teichmüller geodesic trajectories. The main tool is quantitative nondivergence of Teichmüller horocycles, due to Minsky and Weiss. This has an application to billiards in rational polygons.

The compact Hausdorff space X has the CSWP if every subalgebra of C(X, C) which separates points and contains the constant functions is dense in C(X, C). W. Rudin showed that all scattered X have the CSWP. We describe a class of non-scattered X with the CSWP; by another result of Rudin, such X cannot be metrizable.

We prove that the set of all points of effective Hausdorff dimension 1 in Rn (n ≥ 2) is connected, and simultaneously that the complement of this set is not path-connected when n = 2.

In the original version of the present paper submitted in 1995, we stated: "As far as we know it may be possible to prove that, if H and K are two compact Hausdorff spaces and (C(H), pointwise) and (C(K), pointwise) are both a-fragmented, then (C(HxK), pointwise) is also a-fragmented. If this is so, then the condition that for each finite subset O of F the space C(U {Kv: (pe}, pointwise) is ...

There is a locally compact Hausdorff space which is linearly Lindelöf and not Lindelöf. This answers a question of Arhangel’skii and Buzyakova.