Baire Spaces and Quasicontinuous Mappings

Ján Bors\'\i k, \uLubica Holá, Dušan Hol\'y

The notion of quasicontinuity was perhaps the first time used by R. Baire in [2]. Let $X,Y$ be topological spaces and $Q(X,Y)$ be the space of quasicontinuous mappings from $X$ to $Y$. If $X$ is a Baire space and $Y$ is metrizable, in $Q(X,Y)$ we can approach each $(x,y)$ in the graph $Grf$ of $f$ along some trajectory of the form $\{x_k,f_{nk}(xk)):k\in\omega\}$ if and only if we can approach most points along a vertical trajectory. This result generalizes Theorem 5 from [3]. Moreover in the class of topological spaces with the property QP we give a characterization of Baire spaces by the above mentioned fact. We also study topological spaces with the property QP.