Re: a question of Lubarsky

[Thomas Streicher <streicher@mathematik.tu-darmstadt.de>]

> The cardinality argument also shows a bit more, I guess.  As you've point=
> ed out, the valid formulas in Sh(X) and Sh(Y) will agree if they have the=
>  same soberification / locale, or if Y is a product of X with some discre=
> te space.  However, even after identifying such spaces, there will still =
> be (certainly in classical metatheory, and I think in most weaker theorie=
> s) more than continuum-many classes of spaces; so these "trivial reasons"=
>  can't be the only cases when Sh(X) and Sh(Y) validate the same formulas.

Certainly, yust consider algebraic lattices with their Scott topology. They are
all sober and connected. P(\kappa) for all cardinals \kappa provides a class
of nonisomorphic such gadgets.

BTW Bob suggested to characterise when two sober spaces or locales are
logically indistinguishible. I find this a most difficult questions. Reminds me
a bit (admittedly somewhat vague analogy) of characterising elementary
equivalence of structures. There is an answer by Ehrenfeucht-Fraisse games.
But this can't be use for the question at issue.

Thomas