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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)].

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)