On a vector space construction by Hausdorff

Abstract

Initroduction. Some years ago Hausdorff, using Hamel bases, showed that any infinite dimensional real Banach space contained a second category linear subspace that was not complete under any equivalent norm [5]. It is shown below that a slight abstraction of his construction leads to the following: (A) a multiple offender example (Theorem 3), one that combines several pathological features which have been separately discussed in the literature;2 (B) the partitioning of any infinite dimensional second category real linear topological space into a continuum of pairwise disjoint linear manifolds each of which is everywhere dense in the space (a result comparable to the partitionings of certain linear topological spaces into disjoint everywhere dense convex sets by Tukey [15 ] and Klee [8 ]); (C) some miscellaneous observations, including the fact that in some recent topological group theorems certain hypotheses cannot be dropped. To be precise concerning (A), we shall establish the following theorem, in which En is n-dimensional real Euclidean vector space and L. represents, for any element c of En, the set of rational multiples of c. Denoting complements by primes, a set E in En will be called m-thick if both E and E' have zero interior measure; when E is in a topological space X it is c-thick if each of the sets E and E' contains no Baire subsets of X other than first category sets.