Dimension phenomena associated with \(\beta \mathbb N\)-spaces
type: Item
instance of: scholarly article
title: Dimension phenomena associated with \(\beta \mathbb N\)-spaces
zbMATH Open document ID: 1027.54034
DOI: 10.1016/S0166-8641(01)00281-4
author: Ilijas Farah
published in: Topology and its Applications
publication date: 2003-03-10T00:00:00Z
review text: This paper presents some very powerful and striking results concerning the rigidity of powers of the Čech-Stone compactification. The author significantly extends the earlier ground-breaking work of van Douwen. The main result states that if any power of a compact space maps onto a \(\beta {\mathbb N}\)-space, then that power can be partitioned into finitely many clopen pieces so that the map restricts on each piece to one that depends on a single coordinate only. A space is said to be a \(\beta {\mathbb N}\)-space if every countable discrete subset has closure homeomorphic to \(\beta {\mathbb N}\). The author is able to obtain very strong additional mapping properties of \(\beta {\mathbb N}\)-spaces by assuming a principle, wEP, which was shown to follow from OCA plus MA (e.g., PFA) in an earlier related paper. One elegant sample would be that wEP implies that if countable ordinals \(\alpha<\gamma\) are indecomposable, then no power of \(\alpha^*\) maps onto \(\gamma^*\) (where \(X^*\) denotes the Cech-Stone remainder).
reviewed by: Alan Dow
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54D40
zbMATH DE Number: 1879125
zbMATH Keywords: Čech-Stone compactifications
zbMATH Keywords: Weak Extension Principle
zbMATH Keywords: rigidity phenomena
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1879125