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Compositions of simple maps

Jerzy Krzempek — 1999

Fundamenta Mathematicae

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...

Fully closed maps and non-metrizable higher-dimensional Anderson-Choquet continua

Jerzy Krzempek — 2010

Colloquium Mathematicae

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.

Finite-to-one maps and dimension

Jerzy Krzempek — 2004

Fundamenta Mathematicae

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...

On Dimensionsgrad, resolutions, and chainable continua

Michael G. Charalambous;  Jerzy Krzempek — 2010

Fundamenta Mathematicae

For each natural number n ≥ 1 and each pair of ordinals α,β with n ≤ α ≤ β ≤ ω(⁺), where ω(⁺) is the first ordinal of cardinality ⁺, we construct a continuum S n , α , β such that (a) d i m S n , α , β = n ; (b) t r D g S n , α , β = t r D g o S n , α , β = α ; (c) t r i n d S n , α , β = t r I n d ₀ S n , α , β = β ; (d) if β < ω(⁺), 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(⁺), then S n , α , β can be made hereditarily indecomposable; (g) if n = 1 and α = β < ω(⁺), then S n , α , β can be made chainable and hereditarily indecomposable. In particular,...

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