Homeomorphisms of linear and planar sets of the first category into microscopic sets (Q409704)

A set \(A\subset \mathbb{R}\) is \textit{microscopic} if for each \(\varepsilon>0\) there exists a sequence \((I_n)_n\) of intervals such that \(A\subset\bigcup_n I_n\) and \(|I_n|<\varepsilon^n\) for each \(n\). This kind of small sets was introduced by J. Appell. See [\textit{J. Appell, E.~D'Aniello} and \textit{M.~Väth}, Ric. Mat. 50, No. 2, 255--274 (2001; Zbl 1102.28300); Ric. Mat. 54, No. 1, 211--213 (2005; Zbl 1142.26001)]. The authors: (1)~construct an example of Cantor set which is microscopic; (2)~show that each set of the first category on the line can be transformed into a microscopic set by an authomorphism, and such homeomorphisms constitute a residual set in the space of all automorphisms; (3)~show an analogous result for subsets of the plane.