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On exposed points and extremal points of convex sets in ℝⁿ and Hilbert space

Stoyu BarovJan J. Dijkstra — 2016

Fundamenta Mathematicae

Let be a Euclidean space or the Hilbert space ℓ², let k ∈ ℕ with k < dim , and let B be convex and closed in . Let be a collection of linear k-subspaces of . A set C ⊂ is called a -imitation of B if B and C have identical orthogonal projections along every P ∈ . An extremal point of B with respect to the projections under is a point that all closed subsets of B that are -imitations of B have in common. A point x of B is called exposed by if there is a P ∈ such that (x+P) ∩ B = x. In the present...

Homeomorphism groups of Sierpiński carpets and Erdős space

Jan J. DijkstraDave Visser — 2010

Fundamenta Mathematicae

Erdős space is the “rational” Hilbert space, that is, the set of vectors in ℓ² with all coordinates rational. Erdős proved that is one-dimensional and homeomorphic to its own square × , which makes it an important example in dimension theory. Dijkstra and van Mill found topological characterizations of . Let M n + 1 , n ∈ ℕ, be the n-dimensional Menger continuum in n + 1 , also known as the n-dimensional Sierpiński carpet, and let D be a countable dense subset of M n + 1 . We consider the topological group ( M n + 1 , D ) of all...

On closed sets with convex projections in Hilbert space

Stoyu BarovJan J. Dijkstra — 2007

Fundamenta Mathematicae

Let k be a fixed natural number. We show that if C is a closed and nonconvex set in Hilbert space such that the closures of the projections onto all k-hyperplanes (planes with codimension k) are convex and proper, then C must contain a closed copy of Hilbert space. In order to prove this result we introduce for convex closed sets B the set k ( B ) consisting of all points of B that are extremal with respect to projections onto k-hyperplanes. We prove that k ( B ) is precisely the intersection of all k-imitations...

The ambient homeomorphy of certain function and sequence spaces

Jan J. DijkstraJerzy Mogilski — 1996

Commentationes Mathematicae Universitatis Carolinae

In this paper we consider a number of sequence and function spaces that are known to be homeomorphic to the countable product of the linear space σ . The spaces we are interested in have a canonical imbedding in both a topological Hilbert space and a Hilbert cube. It turns out that when we consider these spaces as subsets of a Hilbert cube then there is only one topological type. For imbeddings in the countable product of lines there are two types depending on whether the space is contained in a...

The instability of nonseparable complete Erdős spaces and representations in ℝ-trees

Jan J. DijkstraKirsten I. S. Valkenburg — 2010

Fundamenta Mathematicae

One way to generalize complete Erdős space c is to consider uncountable products of zero-dimensional G δ -subsets of the real line, intersected with an appropriate Banach space. The resulting (nonseparable) complete Erdős spaces can be fully classified by only two cardinal invariants, as done in an earlier paper of the authors together with J. van Mill. As we think this is the correct way to generalize the concept of complete Erdős space to a nonseparable setting, natural questions arise about analogies...

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