Topological properties of some function spaces

Let Y be a metrizable space containing at least two points, and let X be a Y_I-Tychonoff space for some ideal I of compact sets of X. Denote by C_I(X,Y) the space of continuous functions from X to Y endowed with the I-open topology. We prove that C_I(X,Y) is Fréchet–Urysohn iff X has the property γ_I. We characterize zero-dimensional Tychonoff spaces X for which the space C_I(X,2) is sequential. Extending the classical theorems of Gerlits, Nagy and Pytkeev we show that if Y is not compact, then C_p(X,Y) is Fréchet–Urysohn iff it is sequential iff it is a k-space iff X has the property γ. An analogous result is obtained for the space of bounded continuous functions taking values in a metrizable locally convex space. Denote by B₁(X,Y) and B(X,Y) the space of Baire one functions and the space of all Baire functions from X to Y, respectively. If H is a subspace of B(X,Y) containing B₁(X,Y), then H is metrizable iff it is a σ-space iff it has countable cs^⁎-character iff X is countable. If additionally Y is not compact, then H is Fréchet–Urysohn iff it is sequential iff it is a k-space iff it has countable tightness iff X_ℵ0 has the property γ, where X_ℵ0 is the space X with the Baire topology. We show that if X is a Polish space, then the space B₁(X,R) is normal iff X is countable.