# zbMATH — the first resource for mathematics

Baire one star functions. (English) Zbl 0853.54018
The author studies the Baire $${}^*1$$ functions, i.e., functions $$f$$ from a metric space $$(X, \rho)$$ into a metric space $$(Y, \sigma)$$ such that for every nonempty closed subset $$F$$ of $$X$$ there is an open set $$U\subset X$$ intersecting $$F$$ such that the restriction of $$f$$ to $$F$$ is continuous on $$U$$. It is well known that every Baire $${}^*1$$ function is Baire 1 and that for real functions the opposite implication is false.
The paper contains the comparison of the Baire $${}^*1$$ class of functions with the following classes of functions: first Borel class, first level Borel, and piecewise continuous. Then, this comparison is used to prove that the continuous functions from $$(X, \tau)$$ into $$(Y, \eta)$$ are in Baire $${}^*1$$ class for a wide class of topologies $$\tau$$ and $$\eta$$, where $$\tau$$ and $$\eta$$ are finer topologies than the respective metric topologies. In particular the author concludes that the theorem is true for self-maps of $$(\mathbb{R}^n, \tau)$$, $$n\geq 1$$, where $$\tau$$ is an ordinary or strong density topology of a (*-)porosity topology. This last result generalizes similar results for functions from $$\mathbb{R}$$ into $$\mathbb{R}$$ which have appeared in several other papers by the reviewer with L. M. Larson and M. K. Ostaszewski [Forum Math. 2, 265-275 (1990; Zbl 0714.26002); 7, 405-417 (1995; Zbl 0844.26002); $${\mathcal I}$$-density continuous functions, Mem. Am. Math. Soc. 515 (1994; Zbl 0801.26001)].

##### MSC:
 54C99 Maps and general types of topological spaces defined by maps 26A21 Classification of real functions; Baire classification of sets and functions 54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)