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A category analogue of the density topology. (English) Zbl 0613.26002

In the frame (X,S,I), where (X,S) is a measurable space and \(I\subset S\) is a proper \(\sigma\)-ideal, it is possible to generalize, in a quite natural manner, some notions and problems which are fundamental in measure theory and in the theory of continuity [see the authors, Commentat. Math. Univ. Carol. 26, 553-563 (1985; Zbl 0587.54056); E. Wagner, Fundam. Math. 112, 89-102 (1981; Zbl 0449.28006); E. Wagner and W. Wilczyński, Acta Math. Sci. Hung. 36, 125-128 (1980; Zbl 0508.40002)]. The notion of I-density is defined and some useful properties of I-density are established (Theorems 1 and 2). Then, a topological structure, \({\mathcal T}_ I\), is introduced on the real line \((X={\mathbb{R}})\); \({\mathcal T}_ I\) is, by definition, the I-density topology on R. The topological space (\({\mathbb{R}},{\mathcal T}_ I)\) and the class of continuous functions \(f:({\mathbb{R}},{\mathcal T}_ I)\to ({\mathbb{R}},{\mathcal T}_ u)\) are studied (\({\mathcal T}_ u\) means the natural topology on \({\mathbb{R}}).\)
Then, some important results concerning the I-approximately continuous functions are proved; for intance: Theorem 7. The function f has the Baire property if and only if f is I-approximately continuous I - a.e. Theorem 8. If f is I-approximately continuous, then f is of the first class of Baire and has the Darboux property.
Reviewer: O.Costinescu

MSC:

26A21 Classification of real functions; Baire classification of sets and functions
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
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