Decomposing topological spaces into two rigid homeomorphic subspaces
A topological space A is said to be rigid if the only homemorphism of A onto itself is the identity. A topological space X can be rigidly halved if X is the disjoint union of two subsets A and B so that A is homeomorphic with B and each of A and B is rigid. In the 1982 North- Holland Calendar, J. van Mill asked if the set of real numbers could be rigidly halved. \textit{J. van Mill} and \textit{E. Wattel} have shown [Colloq. Math. 50, 95-102 (1985; Zbl 0588.54031)] that the circle can be rigidly halved, but their technique fails for the set of real numbers. Independent affirmative answers to this question about the real numbers were given by the author [Results in mathematical logic, II, Abstr. Am. Math. Soc. 4, 190 (1983)] and by \textit{F. van Engelen} [Topology Appl. 17, 275-285 (1984; Zbl 0538.54011)]. The paper gives the author's proof that the set of real numbers can be rigidly halved. Furthermore, the proof holds for many separable normed vector spaces over the rationals.
scientific article; zbMATH DE number 4119211