Peter F: For that we need -- as we called it in Cats and Alligators -- Wilson space.

What section number? (Not in the index.)

Actually, not the space but the linearly ordered set, most easily defined as the lexicographically ordered subset, W, of sequences with values in {-1, 0, 1} consisting of all those sequences such that for all n a(n) = 0 => a(n+1) = 0 (take a finite word on {-1,1} and pad it out to an infinite sequence by tacking on 0s).

(The parenthetical explanation presumably is meant to apply only to those words containing at least one 0, i.e. W should also contain the infinite words on {-1,1} (else it would be countable).) So shouldn't it be called the Wilson chain then, rather than Wilson space? Ditto for the turkey chain I described yesterday (rationals in quadruplicate, vs. triplicate for the Wilson chain). As a topological space, I don't understand what Wilson space is. The order interval topology (generated by the open order intervals, sets of the form {x | p<x<q}) on a chain makes the irrationals Baire space, the reals the continuum, and the-reals-with-each-rational-duplicated Cantor space. If I haven't miscalculated, it makes the Wilson chain simply C+N, the sum of Cantor space and discrete N (natural numbers). (N arises as the middle rational in each triple. Each middle rational appears simultaneously as the greatest element of one open interval and the least element of another; intersect those to get a singleton open.) If Wilson space is not the order interval topology, i.e. not C+N, then surely it must not be Hausdorff. The only natural possibility I could think of in that case is the 1-point compactification of C iterated omega times, i.e. each element of N appears only in cofinite opens, which didn't seem likely. In any of these cases, my analogously defined "turkey space" looks like it ought to be homeomorphic to Wilson space. Vaughan 28-Nov-2004 13:37:05 -0400,5008;000000000000-00000000