Topological properties of some function spaces

Let $Y$ be a metrizable space containing at least two points, and let $X$ be a $Y_{\mathcal{I}}$-Tychonoff space for some ideal $\mathcal{I}$ of compact sets of $X$. Denote by $C_{\mathcal{I}}(X,Y)$ the space of continuous functions from $X$ to $Y$ endowed with the $\mathcal{I}$-open topology. We prove that $C_{\mathcal{I}}(X,Y)$ is Fr\'{e}chet - Urysohn iff $X$ has the property $\gamma_{\mathcal{I}}$. We characterize zero - dimensional Tychonoff spaces $X$ for which the space $C_{\mathcal{I}}(X,{\bf 2})$ 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 Fr\'{e}chet - 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 Fr\'{e}chet - 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 $B_{1}(X,\mathbb{R})$ is normal iff $X$ is countable.