Bem. (a) dick <=> enthaelt beliebig lange Intervalle
(b) dick synd.
[proof]
Betrachte , ,
1) dick, nicht syndetisch.
2) synd, nicht dick.
(c)
(d) dick (pws, synd) => dick (pws, synd)
synd => synd.
Translation
Chapter 2 Thick subsets of
Definition: thick <=> has the finite intersection property (FIP)
Proposition: TFAE
(a) thick;
(b)
(c) (minimal) left ideal (LID).
[proof of a <=> c]
note:
therefore: .
Remark thick => central.
proof. central => .
Remark.
(a) thick <=> contains arbitrarily long intervals
(b) thick syndetic, thick syndetic.
[proof]
Consider a partition , with
1) thick, not syndetic.
2) syndetic, not thick.
(c)
(d) thick (pws, synd) => thick (pws, synd)
synd => synd.
Notes
Same lecture, new chapter. This is the first of two pages on the basics of thick sets.
“Thick” is an odd notion. It always seems a little made up to me, something stated after the fact (after asking “what does a set look like that covers a left ideal?”). On the other hand, for , I can imagine that the notion “a set that contains arbitrarily long intervals” might actually come up independently of ultrafilters. However, I don’t know the history of the notion, so I’m probably wrong here (if you know anything about this, please leave a comment).
A technical note. I realized that using the section heading “partial translation” was a bit misleading; as would be “augmented/corrected translation”. In fact, I do both – leave some things out (negligible comments etc), clear up the layout, and add corrections (e.g. instead of ). So I will just call it “translation” from now on.
