The embedding structure for linearly ordered topological spaces
type: Item
instance of: scholarly article
title: The embedding structure for linearly ordered topological spaces
zbMATH Open document ID: 1250.54032
published in: Topology and its Applications
publication date: 2012-08-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1102.2159
reviewed by: Jorge Picado
Mathematics Subject Classification ID: 54F05
Mathematics Subject Classification ID: 06F30
Mathematics Subject Classification ID: 03E75
zbMATH DE Number: 6066694
zbMATH Keywords: linearly ordered topological space
zbMATH Keywords: LOTS
zbMATH Keywords: universal structure
zbMATH Keywords: basis
zbMATH Keywords: well-quasi-order
MaRDI profile type: Publication
OpenAlex ID: W2964091957
arXiv ID: 1102.2159
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DOI: 10.1016/J.TOPOL.2012.05.023
author: A. Primavesi
author: K. Thompson
review text: A \textit{linearly ordered topological space} (LOTS) is a linear order endowed with the open interval topology. A \textit{linear order embedding} is an injective order-preserving map and a \textit{LOTS embedding} is an injective order-preserving map that is continuous. The LOTS embeddability relation is a quasi-order on the class of all LOTS.NEWLINENEWLINEThe paper under review shows that this quasi-order in general looks very different from that of linear order embeddability, but that there are some similarities when one restricts to certain subclasses of LOTS. More specifically, the paper treats the following three classes of problems about the LOTS embeddability order:NEWLINENEWLINE(1) Is there a non-trivial set of linear orders such that every linear order in a given class must be LOTS-embeddable into one of them?NEWLINENEWLINE(2) Is there a non-trivial set of linear orders such that every linear order in a given class must LOTS-embed one of them?NEWLINENEWLINE(3) What are the possible cardinalities of chains and antichains in the LOTS embeddability order for a given class of LOTS?NEWLINENEWLINEThe paper provides the following partial answers to the above questions:NEWLINENEWLINE(1) For countable LOTS, the rationals form a universal (i.e. maximal) element for the LOTS embeddability order, but there can be no universal LOTS for any cardinal \(\kappa\) greater than or equal to the size of the continuum; in fact, the dominating number for the LOTS embeddability order is maximal, i.e. \(2^{\kappa}\). There is however a class of linear orders of size \(\kappa\) for which a universal LOTS exists (Section 4).NEWLINENEWLINEBy restricting to certain subclasses of LOTS the authors find similarities between the LOTS embeddability order and that of the linear orders. In Section 5 they prove that universal LOTS for the class of separable partial orders can exist at various cardinals, by way of some general results on dense subsets and continuity. These results generalise to …
description: scientific article; zbMATH DE number 6066694