Reticulated sets and the isomorphism of analytic powers

Document Type

Article

Publication Date

1-1-1985

Abstract

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 nth 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: (1) 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: (2) 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 nth 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. © 1985 by Pacific Journal of Mathematics.

Publication Title

Pacific Journal of Mathematics

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