Peter Krautzberger on the web

Red workbook, p7

Source

red workbook, p7
Red Workbook, p.7

Transcript

  • Kapitel 2 Dicke Teilmengen von $S$
  • Def: $T\subseteq S$ dick <=> ${ x^{-1}T : x\in S }$ hat eDe
  • Prop: Aeq: (a) $T$ dick; (b) $\forall e \subseteq_ e S \exists y \in S: ( y \in \bigcap_ {x\in e} x^{-1} T ) \Longleftrightarrow \stackrel{\stackrel{\text{punktweise}}{\downarrow}}{e \cdot y} \subseteq T)$
    (c) ex. $L \subseteq \widehat{T}$ (min) LID.
    • [proof]
    • a <=> c: $x\in S, p\in \beta S: x^{-1} T \in p \stackrel{\text{Skript}}{\Leftrightarrow} T\in x\cdot p \Leftrightarrow x\cdot p \in \widehat{T}$
    • also $T \text{ dick} \Leftrightarrow \exists p \in \beta S: S\cdot p \subseteq \widehat{T} \stackrel{\text{stetig & closed}}{\Longleftrightarrow} \exists p \in \beta S: \beta S \cdot p \subseteq \widehat{T} \Leftrightarrow \text{Beh.}$
  • Bem. $A$ dick => $A$ zentral.
    • [$A$ dick => $\exists L \text{ min. LID} \subseteq \widehat{A} \Rightarrow \exists \epsilon: E_ \min \cap L \subseteq \widehat{A} \Rightarrow A \in \epsilon$.
  • Bem. (a) $A \subseteq (\omega, +)$ dick <=> $A$ enthaelt beliebig lange Intervalle
    • (b) dick $\stackrel{\not \Rightarrow}{\not \Leftarrow}$ synd.
    • [proof]
      • Betrachte $\omega = \bigcup_ {n \in \omega} I_ n$, $\left\vert I_ n\right\vert = n$, $0\in I_ 0 « I_ 1 « \ldots $
      • 1) $A := \bigcup_ {2 \in \omega} I_ {2n}, B = \bigcup_ {n\in \omega} I_ {2n+1} \rightarrow A, B$ dick, nicht syndetisch.
      • 2) $A = 2 \cdot \mathbb{N}$ synd, $A$ nicht dick.
    • (c) $A \subseteq \text{ pws} \Leftrightarrow \exists g \subseteq_ e S { t^{-1} \bigcup_ {x \in g} x^{-1}A : t \in S} \text{ eDE} \Leftrightarrow \exists g \subseteq_ e : \bigcup_ {x \in g} x^{-1}A \text{ dick.}$
    • (d) $A$ dick (pws, synd) => $x^{-1} A$ dick (pws, synd)
    • $A$ synd => $x\cdot A$ synd.

Translation

  • Chapter 2 Thick subsets of $S$
  • Definition: $T\subseteq S$ thick <=> $ { x^{-1}T : x\in S }$ has the finite intersection property (FIP)
  • Proposition: TFAE
    • (a) $T$ thick;
    • (b) $\forall e \subseteq S \text{ finite } \exists y \in S: ( y \in \bigcap_ {x\in e} x^{-1} T) \Longleftrightarrow \stackrel{\stackrel{\text{pointwise}}{\downarrow}}{e \cdot y} \subseteq T)$
    • (c) $\exists L \subseteq \widehat{T}$ (minimal) left ideal (LID).
    • [proof of a <=> c]
      • note: $x\in S, p\in \beta S: x^{-1} T \in p \Leftrightarrow T\in x\cdot p \Leftrightarrow x\cdot p \in \widehat{T}$
      • therefore: $T \text{ thick} \Leftrightarrow \exists p \in \beta S: S\cdot p \subseteq \widehat{T} \stackrel{\text{continuous & closed}}{\Longleftrightarrow} \exists p \in \beta S: \beta S \cdot p \subseteq \widehat{T} \Leftrightarrow \text{ the claim}$.
  • Remark $A$ thick => $A$ central.
    • proof. $A$ central => $\exists L \text{ min. LID} \subseteq \widehat{A} \Rightarrow \exists \epsilon: E_ \min \cap L \subseteq \widehat{A} \Rightarrow A \in \epsilon$.
  • Remark.
    • (a) $A \subseteq (\omega, +)$ thick <=> $A$ contains arbitrarily long intervals
    • (b) thick $\not \Rightarrow$ syndetic, thick $\not \Leftarrow$ syndetic.
    • [proof]
      • Consider a partition $\omega = \bigcup_ {n \in \omega} I_ n$, $\vert I_ n\vert = n+1$ with $0\in I_ 0 « I_ 1 « \ldots $
      • 1) $A := \bigcup_ {2 \in \omega} I_ {2n}, B = \bigcup_ {n\in \omega} I_ {2n+1} \rightarrow A, B$ thick, not syndetic.
      • 2) $A = 2 \cdot \mathbb{N}$ syndetic, $A$ not thick.
    • (c) $A \subseteq \text{ piecewise syndetic (pws)} \Leftrightarrow \exists g \subseteq S \text{ finite } { t^{-1} \bigcup_ {x \in g} x^{-1}A : t \in S} \text{ eDE} \Leftrightarrow \exists g \subseteq S \text{ finite} : \bigcup_ {x \in g} x^{-1}A \text{ thick.}$
    • (d) $A$ thick (pws, synd) => $x^{-1} A$ thick (pws, synd)
    • $A$ synd => $x\cdot A$ 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 $\omega$, 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. $\vert I_ n\vert = n+1$ instead of $n$). So I will just call it “translation” from now on.