Universal space in the Cartesian product of Peano curves without free arcs
type: Item
instance of: scholarly article
title: Universal space in the Cartesian product of Peano curves without free arcs
zbMATH Open document ID: 1196.54058
author: Daria Michalik
published in: Topology and its Applications
publication date: 2010-05-10T00:00:00Z
review text: A nontrivial continuum \(K\) is a Peano curve (without free arcs) if it is a \(1\)-dimensional locally connected metric compact space (such that no open subset of \(K\) is homeomorphic to an interval). Let \(K_1, \dots, K_{n+1}\), \(n \geq 1,\) be Peano curves without free arcs. The author shows that in the Cartesian product \(K_1 \times \dots \times K_{n+1}\) there exists a universal space \(T_n\) for the class of \(n\)-dimensional separable metric spaces. Moreover, for every separable metric space \(X\) satisfying dim \(X \leq n\) the set of embeddings of \(X\) into \(T_n\) is a residual set in \(C(X, K_1 \times \dots \times K_{n+1}),\) and the set of embeddings of \(X\) into \(K_1 \times \dots \times K_n \times I\), where \(I\) is a closed bounded interval, is a dense set in \(C(X, K_1 \times \dots \times K_n \times I)\). Some additional results concern extensions of mappings. One of them is the following. For a given separable metric space \(X\), a given Peano curve \(K\) without free arcs and a countable family of mappings \(\{f_i : X \to X\}_{i=1}^\infty\) the set of embeddings \(h: X \to K^\omega\) such that there exist extensions \(\{f^*_i : \overline{h(X)} \to \overline{h(X)}\}_{i=1}^\infty\) of the maps \(\{hf_ih^{-1}\}_{i=1}^\infty\) is a residual set in \(C(X, K^\omega).\)
Mathematics Subject Classification ID: 54F45
Mathematics Subject Classification ID: 54C55
Mathematics Subject Classification ID: 54C25
Mathematics Subject Classification ID: 54F50
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 5706279
zbMATH Keywords: covering dimension
zbMATH Keywords: Peano curve
zbMATH Keywords: embedding
zbMATH Keywords: ANR
reviewed by: Vitalij A. Chatyrko
MaRDI profile type: Publication
full work available at URL: j.topol.2010.02.030
OpenAlex ID: W2009987521
cites work: Q4070025
cites work: Q4855460
cites work: Q3994935
cites work: Sur la compactification des espaces métriques
cites work: Dimension preserving compactifications allowing extensions of continuous functions
cites work: Q3304890
cites work: Remark on compactifications of metrizable spaces
cites work: Q4277438
DOI: 10.1016/J.TOPOL.2010.02.030
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description: scientific article; zbMATH DE number 5706279