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Resolving a question of Arkhangel'skiĭ's

100%
Charalambous M.
Fundamenta Mathematicae
|
2006
|
tom 192
|
nr 1
67-76
EN
We construct in ZFC a cosmic space that, despite being the union of countably many metrizable subspaces, has covering dimension equal to 1 and inductive dimensions equal to 2.
2
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The dimension of metrizable subspaces of Eberlein compacta and Eberlein compactifications of metrizable spaces

100%
Charalambous M.
Fundamenta Mathematicae
|
2004
|
tom 182
|
nr 1
41-52
EN
We prove that every Baire subspace Y of c₀(Γ) has a dense $G_δ$ metrizable subspace X with dim X ≤ dim Y. We also prove that the Kimura-Morishita Eberlein compactifications of metrizable spaces preserve large inductive dimension. The proofs rely on new and old results concerning the dimension of uniform spaces.
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On Dimensionsgrad, resolutions, and chainable continua

63%
Charalambous M., Krzempek J.
Fundamenta Mathematicae
|
2010
|
tom 209
|
nr 3
243-265
EN
For each natural number n ≥ 1 and each pair of ordinals α,β with n ≤ α ≤ β ≤ ω(𝔠⁺), where ω(𝔠⁺) is the first ordinal of cardinality 𝔠⁺, we construct a continuum $S_{n,α,β}$ such that (a) $dim S_{n,α,β} = n$; (b) $trDg S_{n,α,β} = trDgo S_{n,α,β} = α$; (c) $trind S_{n,α,β} = trInd₀S_{n,α,β} = β$; (d) if β < ω(𝔠⁺), then $S_{n,α,β}$ is separable and first countable; (e) if n = 1, then $S_{n,α,β}$ can be made chainable or hereditarily decomposable; (f) if α = β < ω(𝔠⁺), then $S_{n,α,β}$ can be made hereditarily indecomposable; (g) if n = 1 and α = β < ω(𝔠⁺), then $S_{n,α,β}$ can be made chainable and hereditarily indecomposable. In particular, we answer the question raised by Chatyrko and Fedorchuk whether every non-degenerate chainable space has Dimensionsgrad equal to 1. Moreover, we establish results that enable us to compute the Dimensionsgrad of a number of spaces constructed by Charalambous, Chatyrko, and Fedorchuk.
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