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.
