We study the properties
of separable measurable spaces which are “Borel-dense of order n.” Those
Borel-dense of order 1 are precisely those that embed as a subset of the unit
interval with totally imperfect complement, and the n-th order version is
an appropriate casting of this idea into n dimensions. The concept enables
one to sharpen some known results concerning the isomorphism types of
analytic spaces. A result of Mauldin and Shortt (separately) may be stated
thus:
If X is a space Borel-dense of order 1 and is Borel-isomorphic with X×X,
then X is automatically a standard (absolute Borel) space. (Mauldin
assumed X to be analytic.)
We obtain the following enlargement:
If X is a space Borel-dense of order n and Xn is Borel-isomorphic with
Xm (some m > n), then X is an analytic space.
The requirement of n-th order density is not overly severe. Complements (in a
standard space) of universally null sets are Borel-dense of every finite order, for
example; the same may be said for complements of sets always of first category
or, more generally, of sets with Marczewski’s property (s0). Statement 2
might therefore be regarded as a criterion whereby to judge which universally
null sets (or sets always of first category, or sets with property (s0)) are
co-analytic. It should also be mentioned, however, that the problem of finding a
particular Borel-dense non-Borel analytic space A for which A2≅A3 is open; it
may be that “analytic” in statement 2 can be strengthened to “standard”.
The relationship between Borel-density and the Blackwell property is also
noted.
