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1
Content available remote

Characterizing metric spaces whose hyperspaces are homeomorphic to ℓ₂

100%
Banakh T., Voytsitskyy R.
Colloquium Mathematicum
|
2008
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tom 113
|
nr 2
223-229
EN
It is shown that the hyperspace $Cld_{H}(X)$ (resp. $Bdd_{H}(X)$) of non-empty closed (resp. closed and bounded) subsets of a metric space (X,d) is homeomorphic to ℓ₂ if and only if the completion X̅ of X is connected and locally connected, X is topologically complete and nowhere locally compact, and each subset (resp. each bounded subset) of X is totally bounded.
2
Content available remote

Zeros of quadratic functionals on non-separable spaces

81%
Banakh T., Plichko A., Zagorodnyuk A.
Colloquium Mathematicum
|
2004
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tom 100
|
nr 1
141-147
EN
We construct non-separable subspaces in the kernel of every quadratic functional on some classes of complex and real Banach spaces.
3
Content available remote

Means on scattered compacta

81%
Banakh T., Bonnet R., Kubis W.
Topological Algebra and its Applications
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2014
|
tom 2
|
nr 1
EN
We prove that a separable Hausdor_ topological space X containing a cocountable subset homeomorphic to [0, ω1] admits no separately continuous mean operation and no diagonally continuous n-mean for n ≥ 2.
4
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On continuous extension of uniformly continuous functions and metrics

81%
Banakh T., Brodskiy N., Stasyuk I., Tymchatyn E.
Colloquium Mathematicum
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2009
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tom 116
|
nr 2
191-202
EN
We prove that there exists a continuous regular, positive homogeneous extension operator for the family of all uniformly continuous bounded real-valued functions whose domains are closed subsets of a bounded metric space (X,d). In particular, this operator preserves Lipschitz functions. A similar result is obtained for partial metrics and ultrametrics.
5
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The Suslinian number and other cardinal invariants of continua

81%
Banakh T., Fedorchuk V., Nikiel J., Tuncali M.
Fundamenta Mathematicae
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2010
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tom 209
|
nr 1
43-57
EN
By the Suslinian number Sln(X) of a continuum X we understand the smallest cardinal number κ such that X contains no disjoint family ℂ of non-degenerate subcontinua of size |ℂ| > κ. For a compact space X, Sln(X) is the smallest Suslinian number of a continuum which contains a homeomorphic copy of X. Our principal result asserts that each compact space X has weight ≤ Sln(X)⁺ and is the limit of an inverse well-ordered spectrum of length ≤ Sln(X)⁺, consisting of compacta with weight ≤ Sln(X) and monotone bonding maps. Moreover, w(X) ≤ Sln(X) if no Sln(X)⁺-Suslin tree exists. This implies that under the Suslin Hypothesis all Suslinian continua are metrizable, which answers a question of Daniel et al. [Canad. Math. Bull. 48 (2005)]. On the other hand, the negation of the Suslin Hypothesis is equivalent to the existence of a hereditarily separable non-metrizable Suslinian continuum. If X is a continuum with $Sln(X) < 2^{ℵ₀}$, then X is 1-dimensional, has rim-weight ≤ Sln(X) and weight w(X) ≥ Sln(X). Our main tool is the inequality w(X) ≤ Sln(X)·w(f(X)) holding for any light map f: X → Y.
6
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Topologically invariant σ-ideals on Euclidean spaces

81%
Banakh T., Morayne M., Rałowski R., Żeberski S.
Fundamenta Mathematicae
|
2015
|
tom 231
|
nr 2
101-112
EN
We study and classify topologically invariant σ-ideals with an analytic base on Euclidean spaces, and evaluate the cardinal characteristics of such ideals.
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