Vol. 90, No. 1, 1980

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Category in function spaces. I

David John Lutzer and Robert Allen McCoy

Vol. 90 (1980), No. 1, 145–168

In this paper we study Baire category in spaces of continuous, real-valued functions equipped with the topology of pointwise convergence. We show that, for normal spaces, the Baire category of Cπ(X) is determined by the Baire category of Cπ(Y ) for certain small subspaces Y of X and that the category of Cπ(X) is intimately related to the existence of winning strategies in a certain topological game Γ(X) played in the space X. We give examples of certain countable regular spaces for which Cπ(X) is a Baire space and we characterize those spaces X for which Cπ(X) has one of the stronger completeness properties, such as pseudocompleteness or Cech-completeness.

Mathematical Subject Classification 2000
Primary: 54C35
Secondary: 46E10
Received: 16 February 1979
Revised: 25 April 1979
Published: 1 September 1980
David John Lutzer
Robert Allen McCoy