Spaces with bases and \(\pi\)-basis of finite rank
type: Item
instance of: scholarly article
title: Spaces with bases and \(\pi\)-basis of finite rank
zbMATH Open document ID: 0611.54001
DOI: 10.1007/BF01157534
author: P. A. Biryukov
published in: Mathematical Notes
publication date: 1985-01-01T00:00:00Z
review text: An antichain is a family \({\mathcal A}\) of sets such that the inclusion \(A\subseteq B\) is false whenever A,B\(\in {\mathcal A}\) and \(A\neq B\). A family \({\mathcal F}\) has finite rank if there is an integer n such that every antichain \({\mathcal A}\subseteq {\mathcal F}\) with \(| {\mathcal A}| =n+1\) satisfies \(\cap {\mathcal A}=\emptyset\); the least such n is called the rank of \({\mathcal F}.\) By an elementary, pleasing argument, the author shows that every space with a \(\pi\)-base of finite rank has a \(\pi\)-base of rank 1. There follows a solution to a problem posed a decade ago by A. V. Arkhangel'skij: If a Hausdorff space X has a base of finite rank and no disjoint open family of cardinality \(\alpha\), then w(X)\(\leq \alpha\); if in addition X is a ccc-space and SH (the Souslin hypothesis) is assumed, then w(X)\(\leq \omega\). A consequence is the theorem of O. Ferster: Assuming SH, every regular, Hausdorff, ccc-space with a base of finite rank is metrizable.
Mathematics Subject Classification ID: 54A25
zbMATH DE Number: 3988217
zbMATH Keywords: weight
zbMATH Keywords: \(\pi \)-weight
zbMATH Keywords: Suslin number
zbMATH Keywords: density
zbMATH Keywords: countable chain condition
zbMATH Keywords: antichain
zbMATH Keywords: \(\pi \)-base of finite rank
zbMATH Keywords: ccc-space
zbMATH Keywords: Souslin hypothesis
reviewed by: W. W. Comfort
MaRDI profile type: Publication
cites work: Q5721695
cites work: Spaces with bases of countable rank
cites work: Countable small rank and cardinal invariants. II
full work available at URL: https://doi.org/10.1007/bf01157534
OpenAlex ID: W2029156293
description: scientific article; zbMATH DE number 3988217