Realizing quasiordered sets by subspaces of `continuum-like' spaces
type: Item
instance of: scholarly article
title: Realizing quasiordered sets by subspaces of `continuum-like' spaces
zbMATH Open document ID: 0951.06003
DOI: 10.1023/A:1006100103089
author: T. B. M. McMaster
author: Aisling E. McCluskey
published in: Order
publication date: 1999-11-25T00:00:00Z
review text: Given an ordered set \(E\) and a topological space \(X\), it is said that \(E\) can be realized within \(X\) if there is an injection \(j\) from \(E\) into the class of (homeomorphism classes of) subspaces of \(X\) such that, for \(x,y\) in \(E\), \(x\leq y\) if and only if \(j(x)\) is homeomorphically embeddable into \(j(y)\). If \(X\) is any \(T_3\) space whose nonempty open sets have the same infinite cardinality \(b\) and which has a dense subset of cardinality \(a\) such that \(b^a=b\), then every poset of cardinality \(b\) can be realized within \(X\). Any quasiordered set on \(2^c\) points (or fewer) whose partially ordered skeleton has \(c\) points (or fewer) is realizable in \(R\).
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 54H10
zbMATH DE Number: 1333345
zbMATH Keywords: realization in a topological space
zbMATH Keywords: ordered set
zbMATH Keywords: quasiordered set
zbMATH Keywords: partially ordered skeleton
reviewed by: Dušan Adnadjević
MaRDI profile type: Publication
full work available at URL: a:1006100103089
OpenAlex ID: W21619196
description: scientific article; zbMATH DE number 1333345