Characterizations of Baire$^ \ast 1$ functions in general settings

Abstract:

Baire* 1 functions from $\left [ {0,1} \right ]$ to $R$ were defined by R. J. O’Malley. For a general topological space $X$, a function $f:X \to R$ will be said to be Baire* 1 if and only if for every nonempty closed subset $H$ of $X$, there is an open set $U$ such that $U \cap H \ne \emptyset$ and $f\left | H \right .$ is continuous on $U$. Several characterizations of Baire* 1 functions are found by altering the well-known Baire 1 characterization: If $H$ is a nonempty closed subset of the domain of $f$, then $f\left | H \right .$ has a point where $f\left | H \right .$ is continuous. These conditions simply replace "closed subset of the preceding characterization with "subset", "countable subset" or "dense-in-itself subset". The relationships of these characterizations are examined with the domain of $f$ being various spaces. The independence of these conditions from the discrete convergence condition described by Á. Császár and M. Laczkovich is discussed.