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1
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Compositions of simple maps

100%
Krzempek J.
Fundamenta Mathematicae
|
1999
|
tom 162
|
nr 2
149-162
EN
A map (= continuous function) is of order ≤ k if each of its point-inverses has at most k elements. Following [4], maps of order ≤ 2 are called simple.  Which maps are compositions of simple closed [open, clopen] maps? How many simple maps are really needed to represent a given map? It is proved herein that every closed map of order ≤ k defined on an n-dimensional metric space is a composition of (n+1)k-1 simple closed maps (with metric domains). This theorem fails to be true for non-metrizable spaces. An appropriate map on a Cantor cube of uncountable weight is described.
2
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An Exactly Two-to-One Map from an Indecomposable Chainable Continuum

100%
Krzempek J.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2004
|
tom 52
|
nr 3
335-339
EN
It is shown that a certain indecomposable chainable continuum is the domain of an exactly two-to-one continuous map. This answers a question of Jo W. Heath.
3
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Finite-to-one maps and dimension

100%
Krzempek J.
Fundamenta Mathematicae
|
2004
|
tom 182
|
nr 2
95-106
EN
It is shown that for every at most k-to-one closed continuous map f from a non-empty n-dimensional metric space X, there exists a closed continuous map g from a zero-dimensional metric space onto X such that the composition f∘g is an at most (n+k)-to-one map. This implies that f is a composition of n+k-1 simple ( = at most two-to-one) closed continuous maps. Stronger conclusions are obtained for maps from Anderson-Choquet spaces and ones that satisfy W. Hurewicz's condition (α). The main tool is a certain extension of the Lebesgue-Čech dimension to finite-to-one closed continuous maps.
4
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Fully closed maps and non-metrizable higher-dimensional Anderson-Choquet continua

100%
Krzempek J.
Colloquium Mathematicum
|
2010
|
tom 120
|
nr 2
201-222
EN
Fedorchuk's fully closed (continuous) maps and resolutions are applied in constructions of non-metrizable higher-dimensional analogues of Anderson, Choquet, and Cook's rigid continua. Certain theorems on dimension-lowering maps are proved for inductive dimensions and fully closed maps from spaces that need not be hereditarily normal, and some of the examples of continua we construct have non-coinciding dimensions.
5
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On Dimensionsgrad, resolutions, and chainable continua

64%
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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