Peter Krautzberger · on the web

Red workbook, p7

Transcript

• Kapitel 2 Dicke Teilmengen von
• Def: dick <=> hat eDe
• Prop: Aeq: (a) dick; (b)
(c) ex. (min) LID.
• [proof]
• a <=> c:
• also
• Bem. dick => zentral.
• [ dick => .
• Bem. (a) dick <=> enthaelt beliebig lange Intervalle
• (b) dick synd.
• [proof]
• Betrachte , , $0\in I_ 0 << I_ 1 << \ldots$
1. dick, nicht syndetisch.
1. 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 $0\in I_ 0 << I_ 1 << \ldots$
1. thick, not syndetic.
1. 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.