Nowhere dense set mappings on the generalized linear continua
type: Item
instance of: scholarly article
title: Nowhere dense set mappings on the generalized linear continua
zbMATH Open document ID: 0727.04002
DOI: 10.1007/BF00383765
author: K. Muthuvel
published in: Order
publication date: 1990-01-01T00:00:00Z
review text: If F maps a set X into its power set P(X) a subset \(A\subseteq X\) is said to be free if \(x\not\in F(Y)\) for any two distinct x, y in A. For infinite cardinals \((=\) initial ordinals) \(\alpha\) the author defines \(R_{\alpha}\) to be the set of all dyadic sequences of digits 0 and 1 of length \(\alpha\) which have a last digit 1. \(C_{\alpha}\) is the set of all dyadic sequences of length \(\alpha\) which contain at least one digit 1 and one digit 0 but which have no last digit 0. Both sets are linearly ordered by the lexicographic order and have the corresponding order topology. For this the author proves: 1) Let \(\alpha\) be a cardinal, \(\alpha '=cf \alpha\). Then there is a nowhere dense set mapping F on \(C_{\alpha}\) [that means: for every \(x\in C_{\alpha}\), F(x) is a nowhere dense subset of \(C_{\alpha}]\) such that the cardinality of any free set for F is at mot \(2^{\alpha '}\). (The case \(\alpha =\alpha '\) is trivial, so this statement is of interest only for singular \(\alpha\).) 2) If GCH is assumed, then there is a nowhere dense set mapping F on \(C_{\alpha}\) such that the cardinality of any free set for F is at most \(\alpha '\). This extends a theorem of \textit{S. H. Hechler} [Isr. J. Math. 11, 231-248 (1972; Zbl 0246.04003)].
reviewed by: Egbert Harzheim
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 06A05
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 03E50
Mathematics Subject Classification ID: 06F30
zbMATH DE Number: 4200200
zbMATH Keywords: dyadic sequences
zbMATH Keywords: lexicographic order
zbMATH Keywords: order topology
zbMATH Keywords: GCH
zbMATH Keywords: nowhere dense set mapping
MaRDI profile type: Publication
cites work: The existence of an everywhere dense independent set
cites work: Proof of a conjecture of B. Ruziewicz
cites work: Beiträge zur Theorie der Ordnungstypen, insbesondere der \(\eta_{\alpha}\)-Mengen
cites work: Directed graphs over topological spaces: some set theoretical aspects
cites work: Free sets for some special set mappings
Recommended article: Large free set
Recommended article: Free sets for some special set mappings
Recommended article: Two consistency results on set mappings
Recommended article: Q5344989
Recommended article: Q5181373
Recommended article: Free sets for set mappings satisfying some intersection conditions
Recommended article: A note on set mappings: an extension of results of Hajnal and Máté under GCH
Recommended article: Set mappings on generalized linear continua
Recommended article: Non-well-foundedness of well-orderable power sets
Recommended article: Infinite Free Set for Small Measure Set Mappings
description: scientific article; zbMATH DE number 4200200