A theorem on planar continua and an application to automorphisms of the field of complex numbers (Q1062097)

It is well known that the identity and the complex conjugation are the only continuous automorphisms of the complex number field \({\mathbb{C}}\) among the \(2^{{\mathfrak c}}\) automorphisms of \({\mathbb{C}}\). (\({\mathbb{C}}\) is provided with its natural topology.) It is an interesting question to give sufficient conditions for an automorphism \(\psi\) of \({\mathbb{C}}\) to be continuous. For example, if \(\psi\) is measurable as a function on \({\mathbb{R}}^ 2\), then \(\psi\) is continuous. \textit{H. Kestelman} [Proc. Lond. Math. Soc., II. Ser. 53, 1-12 (1951; Zbl 0042.393)] has given some sufficient conditions for the continuity of \(\psi\). The present authors prove that if \(\psi\) is an automorphism of \({\mathbb{C}}\) which is bounded on an \(F_{\sigma}\) subset of the plane which has positive inductive dimension, then \(\psi\) is continuous. The proof is based on two interesting propositions. Prop. 1.1: Let \(\psi\) be an automorphism of \({\mathbb{C}}\) which is bounded on a circle or on a line segment. Then \(\psi\) is continuous. Prop. 1.3: Let K be a continuum in the plane which does not lie on a line. Then the set of differences \(K- K=\{w-z|\) w,z\(\in K\}\) contains an open set. - Remark. Prop. 1.1 is related to Kestelman's Theorem 1 (loc. cit.).