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Homeomorphism Groups and the Topologist's Sine Curve

100%
Dijkstra J., Tahri R.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2010
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tom 58
|
nr 3
269-272
EN
It is shown that deleting a point from the topologist's sine curve results in a locally compact connected space whose autohomeomorphism group is not a topological group when equipped with the compact-open topology.
2
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On exposed points and extremal points of convex sets in ℝⁿ and Hilbert space

100%
Barov S., Dijkstra J.
Fundamenta Mathematicae
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2016
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tom 232
|
nr 2
117-129
EN
Let 𝕍 be a Euclidean space or the Hilbert space ℓ², let k ∈ ℕ with k < dim 𝕍, and let B be convex and closed in 𝕍. Let 𝓟 be a collection of linear k-subspaces of 𝕍. A set C ⊂ 𝕍 is called a 𝓟-imitation of B if B and C have identical orthogonal projections along every P ∈ 𝓟. An extremal point of B with respect to the projections under 𝓟 is a point that all closed subsets of B that are 𝓟-imitations of B have in common. A point x of B is called exposed by 𝓟 if there is a P ∈ 𝓟 such that (x+P) ∩ B = {x}. In the present paper we show that all extremal points are limits of sequences of exposed points whenever 𝓟 is open. In addition, we discuss the question whether the exposed points form a $G_{δ}$-set.
3
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On closed sets with convex projections in Hilbert space

100%
Barov S., Dijkstra J.
Fundamenta Mathematicae
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2007
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tom 197
|
nr 1
17-33
EN
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 $𝓔^{k}(B)$ consisting of all points of B that are extremal with respect to projections onto k-hyperplanes. We prove that $𝓔^{k}(B)$ is precisely the intersection of all k-imitations C of B, i.e., closed sets C that have the same projections as B onto all k-hyperplanes. For every closed convex set B in ℓ² with nonempty interior we construct "minimal" k-imitations C, in the sense that $dim(C∖𝓔^{k}(B)) ≤ 0$. Finally, we show that whenever a compact set has convex projections onto all finite-dimensional planes, then it must be convex.
4
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The instability of nonseparable complete Erdős spaces and representations in ℝ-trees

100%
Dijkstra J., Valkenburg K.
Fundamenta Mathematicae
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2010
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tom 207
|
nr 3
197-210
EN
One way to generalize complete Erdős space $𝔈_{c}$ 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 between the behaviour of complete Erdős space and its generalizations. The discovery that $𝔈_{c}$ is unstable, by which we mean that the space is not homeomorphic to its infinite power, by Dijkstra, van Mill, and Steprāns, led to the solution of a series of problems in the literature. In the present paper we prove by a different method that our nonseparable complete Erdős spaces are also unstable. Another application of $𝔈_{c}$ is that it is homeomorphic to the endpoint set of the universal separable ℝ-tree. Our standard models can also be represented as endpoint sets of more general ℝ-trees, but some universality properties are lost
5
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Homeomorphism groups of Sierpiński carpets and Erdős space

100%
Dijkstra J., Visser D.
Fundamenta Mathematicae
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2010
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tom 207
|
nr 1
1-19
EN
Erdős space 𝔈 is the "rational" Hilbert space, that is, the set of vectors in ℓ² with all coordinates rational. Erdős proved that 𝔈 is one-dimensional and homeomorphic to its own square 𝔈 × 𝔈, which makes it an important example in dimension theory. Dijkstra and van Mill found topological characterizations of 𝔈. Let $Mₙ^{n+1}$, n ∈ ℕ, be the n-dimensional Menger continuum in $ℝ^{n+1}$, also known as the n-dimensional Sierpiński carpet, and let D be a countable dense subset of $Mₙ^{n+1}$. We consider the topological group $𝓗(Mₙ^{n+1},D)$ of all autohomeomorphisms of $Mₙ^{n+1}$ that map D onto itself, equipped with the compact-open topology. We show that under some conditions on D the space $𝓗(Mₙ^{n+1},D)$ is homeomorphic to 𝔈 for n ∈ ℕ ∖ {3}.
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