Projections of \(k\)-dimensional subsets of \(\mathbb{R}^n\) onto \(k\)-dimensional planes and irregular subsets of the Grassmanian manifolds (Q1962087)

The main result of the paper states that for every \(k\)-dimensional subset \(X\) of the \(n\)-dimensional Euclidean space \(\mathbb{R}^n\) with a fixed coordinate system in it there is a \(k\)-dimensional coordinate plane \(\alpha\) and a linear projection \(p:\mathbb{R}^n\to\alpha\) such that \(p(X)\) is of the second category in \(\alpha\). This is a generalization over noncompact sets of a result discovered by Nöbeling in the early 30s. A new technique of irregular sets in Grassmannian manifolds used in the paper makes this generalization quite attractive.