Jan van Mill
Brouwer's Dimensionsgrad
Abstract: The first definition of a dimension function is due to Brouwer. He seems to have been put on the right track by Poincaré who proposed to call a continuum n-dimensional, if it can be dissected in separate pieces by one or more (n{-}1)-dimensional continua. Brouwer refined this idea and defined a dimensional invariant intended for topologically complete spaces without isolated points (in the terminology of his days, normal sets in the sense of Fréchet) which he called Dimensionsgrad. There is something very peculiar about this definition, as was pointed out by Urysohn. As a result of this, Brouwer issued a flood of notes and papers correcting his 'mistake'. Brouwer claimed that his Dimensionsgrad could serve as the basis of an equally important theory of dimension as ordinary topological dimension. In this lecture we will show that Brouwer was right for the class of compact spaces, but wrong for the class of topologically complete spaces.
About Jan van Mill
Jan van Mill (1951) is Professor of Mathematics at the Vrije Universiteit (VU) in Amsterdam. Besides being dean of the faculty of Mathematics and being a resident member of the board of academics at the faculty of Mathematics at the VU University, he is also member of the editorial board of the journal "Fundamenta Mathematicae" and managing editor of the "Tbilisi Mathematical Journal".
