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 Frechet-Urysohn iff X has the property gamma(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 Frechet-Urysohn iff it is sequential iff it is a k-space iff X has the property gamma. An analogous result is obtained for the space of bounded continuous functions taking values in a metrizable locally convex space. Denote by B-1(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 B-1(X, Y), then H is metrizable iff it is a sigma-space iff it has countable cs*-character iff X is countable. If additionally Y is not compact, then H is Frechet-Urysohn iff it is sequential iff it is a k-space iff it has countable tightness iff X(aleph 0 )has the property gamma, where X-aleph 0 is the space X with the Baire topology. We show that if X is a Polish space, then the space B-1(X,R) is normal iff X is countable. (C) 2020 Elsevier B.V. All rights reserved.