From: "Dr. Michael Albert" <albert@esther.rad.tju.edu>
Subject: continuum hypothesis
Date: Tue, 8 Feb 2000 20:47:12 -0500
Newsgroups: sci.math
Summary: implies infinite-dimensional spaces whose only finite-dimensional subspaces are countable sets

Dear Friends:

On at least one occasion there was interest in odd-ball
implications of the continuum hypothesis (if anyone
reading this is a specialist in the field let me assure
you I mean odd-ball in the best possible sense :-)).

Anyway, in _Dimension_Theory_ (W. Hurewicz and H. Wallman), p. 25,
"under the hypothesis of the continuum these even exist
infinite-dimensional spaces whose only finite-dimensional
subspaces are countable sets."  The book cites W. Hurewicz,
"Une remarque sur l'hypothese du continu", Fundamenta Mathematicae,
v. 19 (1932) pp 8-9. [hypothese should have an accent grave over
the first e].

For those not familiar with topological dimension theory, the
game is this.  Consider metric spaces with countable basis (separable).
If every point can be separated from every closed set (by the
null set) the space is of dimension 0.  If every point can
be separated from every closed set by a set of dimension 0, then
the space is of dimension 1.  etc.  This captures the idea
that in a 1-d space, one can separate sets by removing points.
In a 2-d space, one can separate sets by removing 1-d curves.
Etc.  

Hope this amuses.

Best wishes,
  Mike