The embedding structure for linearly ordered topological spaces
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 other cardinals, with an appropriate generalisation of the notion of separability.NEWLINENEWLINE(2) Under the Proper Forcing Axiom there is an 11 element basis for uncountable LOTS, and this is the smallest possible. Moreover there is a six element LOTS basis for linear orders that are dense and have only points of countable cofinality/coinitiality (Section 6).NEWLINENEWLINE(3) The LOTS embedding order has large chains and antichains when the LOTS have cardinality at least the continuum (Section 3).
scientific article; zbMATH DE number 6066694
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