Compactness and subsets of ordered sets that meet all maximal chains
type: Item
instance of: scholarly article
title: Compactness and subsets of ordered sets that meet all maximal chains
zbMATH Open document ID: 0555.06001
DOI: 10.1007/BF00565650
author: J. Ginsburg
published in: Order
publication date: 1984-01-01T00:00:00Z
review text: The space \({\mathcal M}(P)\) of maximal chains of a chain complete ordered set P is studied. By \(P_ 1\) and \(P_ 2\) are denoted two particular ordered sets. A characterization of compactness of \({\mathcal M}(P)\) is as follows: \({\mathcal M}(P)\) is compact iff every nonempty open interval of P has a finite cutset and P contains no subset isomorphic to either of the ordered sets \(P_ 1\) or \(P_ 2\). It is proved that the space \({\mathcal M}(P)\) is compact if P satisfies any of the following conditions: (1) P contains no copy of \(P_ 1\) or \(P_ 2\) and for every \(x\in P\) there is a finite subset F of P which is coinitial above x. (2) P contains no proper N and for every \(x\in P\) there is a finite maximal antichain which contains x. (3) P contains no copy of \(P_ 1\) or its dual and all antichains in P are finite.
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 54D30
zbMATH DE Number: 3884220
zbMATH Keywords: forbidden substructures
zbMATH Keywords: space of maximal chains
zbMATH Keywords: chain complete ordered set
zbMATH Keywords: characterization of compactness
zbMATH Keywords: maximal antichain
reviewed by: Dušan Adnadjević
MaRDI profile type: Publication
cites work: Compact Spaces and Spaces of Maximal Complete Subgraphs
cites work: Antichains and Finite Sets that Meet all Maximal Chains
cites work: Maximal chains and antichains
cites work: Finite cutsets and finite antichains
full work available at URL: https://doi.org/10.1007/bf00565650
OpenAlex ID: W1997730101
description: scientific article; zbMATH DE number 3884220