Vol. 48, No. 1, 1973

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Pseudo-completeness and the product of Baire spaces

Jan Aarts and David John Lutzer

Vol. 48 (1973), No. 1, 1–10

The class of pseudo-complete spaces defined by Oxtoby is one of the largest known classes 𝒞 with the property that any member of 𝒞 is a Baire space and 𝒞 is closed under arbitrary products. Furthermore, all of the classical examples of Baire spaces belong to 𝒞. In this paper it is proved that if x ∈𝒞 and if Y is any (quasi-regular) Baire space, then X × Y is a Baire space. The proof is based on the notion of A-embedding which makes it possible to recognize whether a dense subspace of a Baire space is a Baire space in its relative topology. Finally, examples are presented which relate pseudo-completeness to several other types of completeness.

Mathematical Subject Classification 2000
Primary: 54D99
Received: 30 June 1972
Revised: 14 August 1972
Published: 1 September 1973
Jan Aarts
David John Lutzer