A first countable linearly Lindelöf not Lindelöf topological space
type: Item
instance of: scholarly article
title: A first countable linearly Lindelöf not Lindelöf topological space
zbMATH Open document ID: 1226.54024
published in: Topology and its Applications
publication date: 2011-10-10T00:00:00Z
review text: Recall that a space is Lindelöf iff every open cover has a countable subcover. It is linearly Lindelöf iff every increasing open cover has a countable subcover. Under what circumstances must a linearly Lindelöf space be Lindelöf? In particular, must a first countable linearly Lindelöf space be Lindelöf? It is known that a first countable linearly Lindelöf space \(X\) has size \(\leq \mathfrak c\), and that a linearly Lindelöf space with netweight \(< \aleph_{\omega}\) is Lindelöf. Thus, if \(X\) is first countable linearly Lindelöf but not Lindelöf then \(\mathfrak c > \aleph_{\omega}\) and \(nw(X) \geq \aleph_{\omega}\). The main result of this paper is to show that under \(MA + \aleph_{\omega} < \mathfrak c\) there is a first countable linearly Lindelöf space which is not Lindelöf. The space is not difficult to describe, so we describe it here. Begin with the usual topology on the Cantor set \((C, \tau)\), together with a descending chain mod finite \(\{A_{\alpha}: \alpha < \aleph_{\omega}\} \subset \wp(\omega)\). Define \(\tau_n\) to be the topology on \(C\) generated by \(\tau\) together with all sets of the form \(U_S = \{x \in C: \exists \alpha < \aleph_n \;S =^* A_{\alpha}\) and \(\forall n \in S \;x(n) = 0\}\). Define \(C_n = \bigcup \{U_S: \exists \alpha < \aleph_n \;S =^* A_{\alpha}\}\) and \(\tilde C_n =\) the \(\tau\)-closure of \(C_n\). Then the desired space is \(P = \bigcup_{k < \omega} \prod_{i = 0}^{k}\tilde C_i \times \prod_{i = k+1}^{\infty}C_i\). Assuming the existence of \(\{A_{\alpha}: \alpha < \aleph_{\omega}\}\), \(P\) is first countable and not Lindelöf. That it is linearly Lindelöf depends on more of \(MA + \aleph_{\omega} < \mathfrak c\), but, as the author points out, the full strength of the hypothesis is not used. Another result is that, again under \(MA\), there is a first countable Michael space which refines the usual topology on the Cantor set. This example is obtained by tweaking an example of \textit{K. Alster} [Proc. Am. Math. So…
reviewed by: Judith Roitman
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54A10
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 54G20
zbMATH DE Number: 5956069
zbMATH Keywords: linearly Lindelöf
zbMATH Keywords: Lindelöf
zbMATH Keywords: first countable
author: O. Pavlov
MaRDI profile type: Publication
full work available at URL: j.topol.2011.06.057
OpenAlex ID: W2036533521
cites work: The Product of a Lindelof Space with the Space of Irrationals Under Martin's Axiom
cites work: On linearly Lindelöf and strongly discretely Lindelöf spaces
cites work: Q4955941
cites work: Q4519554
cites work: Function spaces in the topology of pointwise convergence, and compact sets
cites work: Cardinalities of some Lindelöf and \(\omega_1\)-Lindelöf \(T_1/T_2\)-spaces
cites work: Q3994935
cites work: Q4372664
cites work: Sequentially linearly Lindelöf spaces
cites work: Q3024370
cites work: Q3368456
cites work: Q5732306
cites work: Some of the combinatorics related to Michael’s problem
DOI: 10.1016/J.TOPOL.2011.06.057
description: scientific article; zbMATH DE number 5956069