A first-order expansion of the R-vector space structure on R does not define every compact subset of every Rn if and only if topological and Hausdorff dimension coincide on all closed definable sets. Equivalently, if A ⊆ Rk is closed and the Hausdorff dimension of A exceeds the topological dimension of A, then every compact subset of every Rn can be constructed from A using finitely many boolea...

Shape theory is an extension of homotopy theory which uses the idea of homotopy in its conception. By comparison, the theory of n-shape, which heretofore only has been defined for metrizable compacta, has as its basic notion that of n-homotopy instead of homotopy. We shall demonstrate that the theory of n-shape extends to the class of all Hausdorff compacta.

We classify all one-point order-compactifications of a noncompact locally compact order-Hausdorff ordered topological space X. We give a necessary and sufficient condition for a one-point order-compactification of X to be a Priestley space. We show that although among the one-point order-compactifications of X there may not be a least one, there always is a largest one, which coincides with the...

Let us consider two compact connected and locally connected Hausdorff spaces M , N and two continuous functions φ : M → R, ψ : N → R . In this paper we introduce new pseudodistances between pairs (M,φ) and (N,ψ) associated with reparametrization invariant seminorms. We study the pseudodistance associated with the seminorm ‖φ‖ = maxφ − minφ, denoted by δΛ, and we find a sharp lower bound for it....

The aim of the paper is to prove that if L is a linear subspace of the space C(K) of all real-valued continuous functions defined on a nonempty compact Hausdorff space K such that min(|f |, 1) ∈ L whenever f ∈ L, then for any nonzero g ∈ L̄ (where L̄ denotes the uniform closure of L in C(K)) and for any sequence (bn)n=1 of positive numbers satisfying the relation P∞ n=1 bn = ‖g‖ there exists a se...

We study the problem of when the hyperspace CL(A) of a subspace A of a space X is canonically representable as a subspace of the hyperspace CL(X), where both CL(A) and CL(X) are endowed with one of the following hypertopologies: Fell, Wijsman, d-proximal, Hausdorff, locally finite, proximal

Given Banach spaces X, Y and a compact Hausdorff space K, we use polymeasures to give necessary conditions for a multilinear operator from C(K, X) into Y to be completely continuous (resp. unconditionally converging). We deduce necessary and sufficient conditions for X to have the Schur property (resp. to contain no copy of c0), and for K to be scattered. This extends results concerning linear ...

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.

We prove several reflection theorems on D-spaces, which are Hausdorff topological spaces X in which for every open neighbourhood assignment U there is a closed discrete subspace D such that

Let D and A be unital and separable C∗-algebras; let D be strongly selfabsorbing. It is known that any two unital ∗-homomorphisms from D to A ⊗ D are approximately unitarily equivalent. We show that, if D is also K1-injective, they are even asymptotically unitarily equivalent. This in particular implies that any unital endomorphism of D is asymptotically inner. Moreover, the space of automorphi...