Red workbook, p6

Source

Transcript

• Folgerung: Ann. 1_S ex., X = 2^S.

  • Prop. C\subseteq x = \chi_C \in X:
    • C zentral <=> ex y \in X: y prox. x, unif. rek, y(1_S) = q
      • ["=>" wie Satz/Beweis; "<=" Setze U = { z: z(1_Q) = 1} \in y
      • \underset{Bsp.2}{\Rightarrow} R(x,U) = C \stackrel{\text{Def}}{\underset{\text{Satz}}{\Rightarrow}} C zentral]

• S = (\omega, +)

• Wir wissen

  • (a) x,y \in X prox. <=> x,y stimmen auf bel. langen Int. von \omega ueberein
  • (b) y \in X unif. rekurrent \Longleftrightarrow \forall U \in \mathfrak{U}(y): R(y,U) synd.
    • \Longleftrightarrow \forall U_N \text{ (Basis offen) } R(y,U_n) syndetisch
    • \Longleftrightarrow \forall n \exists b \forall l \exists k (l \leq k \leq l+b \text{ und } y \upharpoonright [k,k+n] = y \upharpoonright n

• Fuer C \subseteq \omega:

  • C zentral \Leftrightarrow \exists y \in 2^\omega: y unif. rek. und y prox. $\chi_C $
  • und y(0)=1

• Also zentrale Mengen analytisch!

• Frage: echt analytisch? SK: wahrscheinlich.

• Notiz: Rand Oben: ?: Dieses DS in irgendeinem Sinn universell?

• Notiz: gegenueberliegende Seite:

  • anders: (offenbar ist) die eigentliche Def. dadurch verbessert, dass klar wird, dass Obermengen wieder zentral sind
  • Frage: kann ich im selben DS bleiben, um die Def. fuer eine Obermenge zu verifizieren?
    • orig/anders: Laesst sich die Konstruktion in 2^Q hinueberretten durch eine geeignete Abbildung?

partial Translation

• Conclusion: Assume an identity 1_S exists, X = 2^S.

  • (Side note: is this dynamical system universal in some sense?)
  • Proposition. C\subseteq x = \chi_C \in X:
    • C central <=> ex y \in X: y proximal x, uniformly recurrent, y(1_S) = q
      • ["=>" as in the proof of the theorem; "<=" Let U = { z: z(1_Q) = 1} \in y
      • \underset{Example 2}{\Rightarrow} R(x,U) = C \stackrel{\text{Definition}}{\underset{\text{Theorem}}{\Rightarrow}} C central]

• S = (\omega, +)

• We know

  • (a) x,y \in X proximal <=> x,y agree on arbitrarily long Intervals of \omega
  • (b) y \in X unif. recurrent \Longleftrightarrow \forall U \in \mathfrak{U}(y): R(y,U) syndetic
    • \Longleftrightarrow \forall U_N \text{ (basic open) } R(y,U_n) syndetic
    • \Longleftrightarrow \forall n \exists b \forall l \exists k (l \leq k \leq l+b \text{ und } y \upharpoonright [k,k+n] = y \upharpoonright n

• For C \subseteq \omega:

  • C central \Leftrightarrow \exists y \in 2^\omega: y unif. rec. and y prox. $\chi_C $
    • and y(0)=1

• So central Sets are analytic!

• Question: properly analytic? Sabine Koppelberg: probably.

• Note (across the page)

  • to put it differenlty: this clearly improved the initial Definition as it is now clear that supersets of central sets are central.
  • Question: can we stay in the same dynamical system to verify that supersets of a central set are central?
    • to put it differently: can we find a map to transfer the constructions in 2^Q over to an arbitrary DS?

Notes

This is a curious (double) page. I don't know why Sabine Koppelberg thought it was important to add the observation that the set of central sets of \omega is analytic (and I'm wondering if the question about "properly analytic" came from my PhD-sibling Gido Scharfenberger-Fabian). I don't think I've ever seen this fact used in the wild. It is a nice observation though and lets me add the first entry in this transcription project with the following subsection:

Open problems

• Is the set of central subsets of \omega properly analytic? (Sabine Koppelberg was apparently leaning towards "yes!".)