Dimension phenomena associated with \(\beta \mathbb N\)-spaces
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).
scientific article; zbMATH DE number 1879125