Eighty years ago, Felix Hausdorff and Paul Alexandroff published independently a theorem asserting that every compact metric space is a continuous image of the Cantor set. This theorem found its application in various branches of mathematics and played also an important role in the theory of curves. The complete characterization of continuous interval images (i.e. Jordan curves) given by the Ha...

Let W denote d-dimensional Brownian motion. We find an explicit formula for the essential supremum of Hausdorff dimension of W (E)∩F , where E ⊂ (0 ,∞) and F ⊂ R are arbitrary nonrandom compact sets. Our formula is related intimately to the thermal capacity of Watson (1978). We prove also that when d ≥ 2, our formula can be described in terms of the Hausdorff dimension of E × F , where E × F is...

We study the extent to which the Hausdorff dimension of a compact subset of an infinite-dimensional Banach space is affected by a typical mapping into a finite-dimensional space. It is possible that the dimension drops under all such mappings, but the amount by which it typically drops is controlled by the ‘thickness exponent’ of the set, which was defined in [13]. More precisely, let X be a co...

A sequential space (X,T) is called minimal sequential if no sequential topology on X is strictly weaker than T . This paper begins the study of minimal sequential Hausdorff spaces. Characterizations ofminimal sequential Hausdorff spaces are obtained using filter bases, sequences, and functions satisfying certain graph conditions. Relationships between this class of spaces and other classes of s...

Let A be a given compact subset of the euclidean space. We consider the problem of finding a compact connected set S of minimal 1dimensional Hausdorff measure, among all compact connected sets containing A. We prove that when A is a finite set any minimizer is a finite tree with straight edges, thus recovery the classical Steiner Problem. Analogously, in the case when A is countable, we prove t...

We prove that the topology of a compact Hausdorff topological Heyting algebra is a Stone topology. It then follows from known results that a Heyting algebra is profinite iff it admits a compact Hausdorff topology that makes it a compact Hausdorff topological Heyting algebra.

The main purpose of this paper is to prove some theorems concerning inverse systems and limits of continuous images of arcs. In particular, we shall prove that if X = {Xa, pab, A} is an inverse system of continuous images of arcs with monotone bonding mappings such that cf(card(A)) = ω1, then X = limX is a continuous image of an arc if and only if each proper subsystem {Xa, pab, B} of X with cf...

If an w-bounded group G acts continuously on a compact Hausdorff space X and the orbit of every point is dense in X , then X iscoabsolute to a Cantor cube. A topological group G acts continuously on the topological space X if G is a group of homeomorphisms of X and the natural map X x G —> X is continuous. X is always assumed to be a compact Hausdorff space. We say that G acts minimally on X if...

The paper investigates the variation of the spectrum of operators in infinite dimensional Banach spaces. Consider the space of bounded operators on a separable Banach space when equipped with the strong operator topology, and the Polish space of compact subsets of the closed unit disc of the complex plane when equipped with the Hausdorff topology. Then, it is shown that the unit spectrum functi...

We continue from “part I” our address of the following situation. For a Tychonoff space Y, the “second epi-topology” σ is a certain topology on C(Y), which has arisen from the theory of categorical epimorphisms in a category of lattice-ordered groups. The topology σ is always Hausdorff, and σ interacts with the point-wise addition + on C(Y) as: inversion is a homeomorphism and + is separately c...