On Souslin Sets in the \Sigma-Product in N^\aleph₁
by
J. Chaber
Coauthors: G. Gruenhage, R. Pol

The results discussed here have been obtained jointly with G. Gruenhage and R. Pol.

Let us recall that A subset X is a Souslin set in X if A is the projection of a Borel set in X ×N^N.

Theorem 1 If A is an uncountable Souslin set in \Sigma(N^\aleph₁), then A satisfies one of the following conditions:

1. A contains a Cantor set,
2. A contains a copy of the countable ordinals \omega₁,
3. A can be well-ordered in type \omega₁ so that all segments { v : v < u } are closed in A.

Theorem 1 can be refined in terms of a natural partial order on \Sigma(N^\aleph₁). For u in \Sigma(N^\aleph₁), let \kappa(u) = min{ \alpha: supp(u) subset [0, \alpha) } and define u \preceq v if u|\kappa(u) subset v|\kappa(v). The sets \Sigma_\xi = { u in \Sigma(N^\aleph₁) : \kappa(u) = \xi} will be called the layers of \Sigma(N^\aleph₁).

Theorem 2 If A is an uncountable Souslin set in \Sigma(N^\aleph₁), then A satisfies either condition (c) from Theorem 1, or one of the following two conditions:

1. A contains an \preceq-antichain intersecting all but non-stationary many layers \Sigma_\xi in a Cantor set,
2. A contains a \preceq-chain intersecting all but non-stationary many layers \Sigma_\xi.

It can be shown that condition 3 of Theorem 1 implies that A intersects only non-stationary many layers \Sigma_\xi. Thus, by Theorem 2, selecting a singleton from each layer \Sigma_\xi, we cannot get a Souslin set in \Sigma(N^\aleph₁), unless (neglecting non-stationary many choices) we actually define a \preceq-chain.

Let B(\aleph₁) denote the Baire space of weight \aleph₁, i.e. the space of sequences of countable ordinals with the ``first difference'' metric. The space B(\aleph₁) has a natural stratification into layers B_\xi (generated by \kappa(x) = min{ \alpha: rg(x) subset [0, \alpha)}) and there is a natural embedding of B(\aleph₁) into \Sigma(N^\aleph₁) which preserves the limit layers. Therefore, our result generalizes the known fact that choosing, for each limit \xi, a point x_\xi in B_\xi, we cannot get a Souslin set in B(\aleph₁). A.H.Stone pointed out that this reflects non-effectiveness of any such choice, and our observation concerning selectors for the layers \Sigma_\xi, further supports this point of view.

The above results are related to some classical topics concerning Lusin's constituents.

Our results imply that N^\aleph₁ does not have even very weak covering properties. This answers a question of N.Kemoto and Y.Yajima.

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # caah-12.