Sequentially linearly Lindelöf spaces
type: Item
instance of: scholarly article
title: Sequentially linearly Lindelöf spaces
zbMATH Open document ID: 1022.54013
DOI: 10.1016/S0166-8641(02)00108-6
author: Victoria Lubitch
author: Menachem Kojman
published in: Topology and its Applications
publication date: 2003-05-20T00:00:00Z
review text: A Hausdorff space is called sequentially linearly Lindelöf if for every uncountable regular cardinal \( \kappa\leq w(X)\) and every \(A \subseteq X\) of cardinality \(\kappa\) there exists \(B \subseteq A\) of cardinality \(\kappa\) which converges to a point. The authors prove that the existence of a good \((\mu, \lambda)\)-scale for a singular cardinal \(\mu\) of countable cofinality and a regular \(\lambda > \mu \) implies the existence of a sequentially linearly Lindelöf space of cardinality \(\lambda\) and weight \( \mu\) which is not Lindelöf. They deduce that it is consistent to have linearly Lindelöf non-Lindelöf spaces below the continuum. Also that it is consistent to have a realcompact linearly Lindelöf non-Lindelöf space below \(2^{\aleph_\omega}\). Further that it is consistent to have a Dowker topology on \(\aleph_{\omega +1}\) in which every subset of cardinality \(\aleph_n\), \(n > 0\) has a converging subset of the same cardinality. Furthermore they show that the non-existence of sequentially linearly Lindelöf non-Lindelöf spaces implies the consistency of large cardinals.
reviewed by: T. Thrivikraman
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 03E55
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 54A25
zbMATH DE Number: 1913862
zbMATH Keywords: linearly Lindelöf spaces
zbMATH Keywords: PCF theory
zbMATH Keywords: realcompact spaces
zbMATH Keywords: complete accumulation
zbMATH Keywords: singular cardinals
zbMATH Keywords: large cardinals
MaRDI profile type: Publication
cites work: Q4955941
cites work: On linearly Lindelöf and strongly discretely Lindelöf spaces
cites work: Shelah's pcf theory and its applications
cites work: Squares, scales and stationary reflection
cites work: A ZFC Dowker space in ℵ_{𝜔+1}: An application of pcf theory to topology
cites work: Q3994935
cites work: Q4258066
cites work: Q3024370
cites work: Chang's conjecture for \(\aleph_\omega\)
cites work: Q5732306
cites work: Q4054359
cites work: Sequentially Compact, Franklin-Rajagopalan Spaces
cites work: A normal space X for which X×I is not normal
cites work: Square in core models
cites work: Q4326402
cites work: Q3041184
cites work: Q4824883
cites work: Q4164026
description: scientific article; zbMATH DE number 1913862