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A Note on Small Baire Spaces

Published online by Cambridge University Press:  20 November 2018

Saharon Shelah
Affiliation:
Hebrew University of Jerusalem, Jerusulam, Israel
Stevo Todorcevic
Affiliation:
University of California, Berkeley, Berkeley, California
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A Baire space is a topological space which satisfies the Baire Category Theorem, i.e., in which the intersection of countably many dense open sets is dense. In this note we shall be interested in the size of Baire spaces, so to avoid trivialities we shall consider only non-atomic spaces, that is, spaces X whose regular open algebras ro(X) are non-atomic. All natural examples of Baire spaces, such as complete metric spaces or compact spaces, seem to have sizes at least 20. So a natural question, asked first by W. Fleissner and K. Kunen [5], is whether there exists a Baire space of the minimal possible size ℵ1. The purpose of this note is to show that such a space need not exist by proving the following result.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Abraham, U. and Shelah, S., Forcing closed unbounded sets, J. Symb. Logic 48 (1983), 643657.Google Scholar
2. Baumgartner, J. E., Iterated forcing, Surveys in Set Theory (Cambridge Univ. Press, 1983), 159.Google Scholar
3. Davies, P. C., Small Baire spaces and σ-dense partial orders, Ph.D. thesis, University of Toronto (1979).Google Scholar
4. Easton, W. B., Powers of regular cardinals, Ann. Math. Logic 1 (1970), 139178.Google Scholar
5. Fleissner, W. G. and Kunen, K., Barely Baire spaces, Fund. Math. 101 (1978), 229240.Google Scholar
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8. Shelah, S., Proper forcing, Lecture Notes in Math. 940 (Springer Verlag, Berlin, 1982).CrossRefGoogle Scholar
9. Todorcevic, S., Some consequences of MA + ┐wKH, Top. and Appl. 12 (1981), 187202.Google Scholar
10. Ulam, S., Zur Masstheorie in der algemeinen Mengenlehre, Fund. Math 16 (1930), 140150.Google Scholar
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