Abstract
The following result is due to S. Piccard ([12], S.30): “If A,B ⊂ℝ are Baire sets of second category and if the function f: ℝ×ℝ→ℝ is defined by f(x,y):=x−y (x,y ε ℝ), then the interior of f(A×B) is non void”. In this note the two main results assure, that the theorem of S. Piccard remains valid, if (1) ℝ is replaced by topological spaces X,Y,Z, (2) f:X×Y→Z is a function, which satisfies a certain global (respectively local) solvability condition, (3) A ⊂X contains a Baire set of second category and (4) B ⊂Y is only of second category.
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Sander, W. Verallgemeinerungen eines Satzes von S. Piccard. Manuscripta Math 16, 11–25 (1975). https://doi.org/10.1007/BF01169060
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DOI: https://doi.org/10.1007/BF01169060