In this note we obtain necessary and sufficient conditions for a convergence space to have a smallest Hausdorff compactification and to have a smallest regular compactification. Introduction. A Hausdorff convergence space as defined in [1] always has a Stone-Cech compactification which can be obtained by a slight modification of the result in [3]. But in general this need not be the largest Hau...

In [2] the pseudo-metric distmax min on compact subsets A and B of a topological space generated from arbitrary metric space is defined. Using this notion we define the Hausdorff distance (see e.g. [6]) of A and B as a maximum of the two pseudo-distances: from A to B and from B to A. We justify its distance properties. At the end we define some special notions which enable to apply the Hausdorf...

We prove that a compact metric space (or more generally an analytic subset of a complete separable metric space) of Hausdorff dimension bigger than k can be always mapped onto a k-dimensional cube by a Lipschitz map. We also show that this does not hold for arbitrary separable metric spaces. As an application we essentially answer a question of Urbański by showing that the transfinite Hausdorff...

and Applied Analysis 3 where Hα is a Γ-open set, Gγ is a Σ-open set, and Mγ is a Γ-compact set. It is easy to see that {Gγ ×{0}}γ∈Ω is an open cover ofK×{0}, thus there is a finite subcoverGγ1 ×{0}, . . . , Gγn ×{0}. Then, Gγ1 × {0, 1} \Mγ1 × {1} ∪ · · · ∪Gγn × {0, 1} \Mγn × {1} 1.6 misses only finitely many Γ-compact sets Mγ1 × {1}, . . . ,Mγn × {1}. AsMγj j 1, 2, . . . , n is compact, we have...

It is shown that for every ε ∈ (0, 1), every compact metric space (X, d) has a compact subset S ⊆ X that embeds into an ultrametric space with distortion O(1/ε), and dimH(S) > (1− ε) dimH(X), where dimH(·) denotes Hausdorff dimension. The above O(1/ε) distortion estimate is shown to be sharp via a construction based on sequences of expander graphs.

We define here e-chaos and study its interrelationships with P -chaos and Ruelle-Takens chaos. We show that a continuous self map f on a compact metric space is e-chaotic (P -chaotic) need not imply the induced map f̄ : K(X) → K(X) is e-chaotic (P -chaotic) and vice-versa, where K(X) is the space of all non-empty compact subsets of X endowed with Hausdorff metric. Mathematics Subject Classificat...

We construct a locally compact groupoid with the properties in the title. Our example is based closely on constructions used by Higson, Lafforgue, and Skandalis in their work on counterexamples to the BaumConnes conjecture. It is a bundle of countable groups over the one point compactification of the natural numbers, and is Hausdorff, second countable and étale with compact unit space.

Monotonically metacompact spaces were recently introduced as an extension of the concept of monotonically compact spaces. In this note we answer a question of Popvassilev, and Bennett, Hart, and Lutzer, by showing that every compact, Hausdorff, monotonically (countably) metacompact space is metrizable. We also show that certain countable spaces fail to be monotonically (countably) metacompact.

We show that the Hausdorff dimension of the limit set is a real analytic function on the deformation space of a certain class of convex co-compact Kleinian groups, which includes all convex co-compact function groups. This extends a result of Ruelle [21] for quasifuchsian groups.

Making an extensive use of small transfinite topological dimension trind, we ascribe to every metric space X an ordinal number (or −1 or Ω) tHD(X), and we call it the transfinite Hausdorff dimension of X. This ordinal number shares many common features with Hausdorff dimension. It is monotone with respect to subspaces, it is invariant under bi-Lipschitz maps (but in general not under homeomorph...