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Comparison of \(\psi\)-density topologies. (English) Zbl 1016.26001

Summary: The paper includes a necessary and sufficient condition under which two \(\psi\)-density topologies generated by two functions \(\psi_1\) and \(\psi_2\) are equal. The condition is formulated in terms of the behavior of two sequences of sets \[ A^+_k= \{x\in \mathbb{R}_+: \psi_1(2x)<\textstyle{{1\over k}}\psi_2(2x)\} \] and \[ B^+_k= \{x\in \mathbb{R}_+: \psi_2(2x)< \textstyle{{1\over k}}\psi_1(2x)\}. \] {}.

MSC:

26A03 Foundations: limits and generalizations, elementary topology of the line
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
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