On the first class of Baire generated by continuous functions on \(R^ N\) relative to the almost Euclidean topology (Q1187260)

Let \(A\) denote the class of ordinarily approximately continuous functions \(f:R^ N\to R\) and let \(P\) denote the class of almost everywhere continuous functions. If \(F\) is a class of functions, then \(B_ 1(F)\) denotes the class of functions which are pointwise limits of sequences of functions in \(F\). The reviewer [Colloq. Math. 38, 259-262 (1978; Zbl 0385.26006)] has defined, for \(N=1\), a class of functions \((AP_ 1)\) such that \(B_ 1(A\cap P)\subset B_ 1(A)\cap B_ 1(P)\cap(AP_ 1)\). In this paper the closely related classes \((AP^*_ 1)\) and \((AP_ 1^{**})\) are introduced and the equalities \(B_ 1(A\cap P)=B_ 1(A)\cap(AP_ 1^{**})\) and \((AP_ 1^*)=(AP_ 1)\) are proved.