Red workbook, p13

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First page

• 14. Sept. 2006
• Fortsetzung Vortrag SK
• 3. collectionwise thick (cwt, cwdick), collectionwise pws (cwpws)
• 3.2 Notation. \mathfrak{B} \subseteq \mathcal{P}(S), \mathfrak{V} := \{ D_e : e \in [\mathcal{B}]^{\lt \omega} \}
  • \mathfrak{D} (-Suetterlin?), D_e = \bigcap_{B\in e} B
  • Also: \mathcal{B} \subseteq \mathfrak{V}
• 3.1 Definition
  • Fuer \mathfrak{A} \subseteq \mathcal{P}(S): Y_ {\mathfrak{A}} = \bigcap_ {A\in\mathfrak{A}} \widehat{A} \subseteq \beta S abgeschlossen
    • [in 3.2, dann Y_ \mathfrak{B} = Y_ \mathfrak{D}]
• 3.3 Satz & Def. Aequivalent:
  • (a) \exists q \in \beta S: \beta S \cdot q \subseteq Y_ \mathfrak{B} (natuerlich oBdA q\in K(\beta S))
  • (b) \forall D \in \mathfrak{D} D dick.
  • Dann: heisst \mathfrak{B} cwdick (cwd)
• Beweis
  • =>: \beta S \cdot q \subseteq Y_ {\mathfrak{B}} = Y_ \mathfrak{D} \subseteq \widehat{D} fuer all D\in \mathfrak{D}
  • <=: Fuer e\in [\mathfrak{B}]^{\lt \omega} nimm q_ e \in \beta S, \beta S \cdot q_ e \subseteq \widehat{D_ e}
    • OBdA, q\in K(\beta S), (q_e \in \beta S \cdot q_e \in \widehat{D_e} gilt ✓)
    • Setze X_e := \{ q_f : e \subseteq f \in [ \mathfrak{B} ]^{\lt \omega} \} \subseteq \beta S.

Second page

• Damit X_e \subseteq \widehat{D_e} [ f\supseteq e \Rightarrow q_f \in \widehat{D_f} \subseteq \widehat{D_e}]
• \{ X_e : e \in [\mathfrak{B}]^{\lt \omega} \} hat eDE (? check)
• nimm q\in \bigcap_{e \in [ \mathfrak{B} ]^{\lt \omega}} cl_{\beta S}(X_e)
• Beh. \forall D \in \mathfrak{D}: \beta S \cdot q \subseteq \widehat{D}
  • [ D = D_e, e\in [ \mathfrak{B}]^{\lt \omega} \Rightarrow X_e \subseteq \widehat{D_e}, q\in cl(X_e) \Rightarrow q \in D_e
  • \forall S \in S, e \subseteq f, s\cdot q_f \underset{\beta S \cdot q_f \subseteq\widehat{D_f}}{\in} \widehat{D_f} \subseteq \widehat{D_e} \Rightarrow s\cdot X_e \subseteq \widehat{D_e}
  • \Rightarrow s\cdot q \in \widehat{D_e} \Rightarrow \beta S \cdot q \subseteq \widehat{D_e}]
  • Damit folgt die Behauptung □
• ["Aufgabe": konstruiere q durch p-limiten?]
• 3.4 Satz & Def. Aequivalent
  • (a) \exists q\in K(\beta S) mit \mathfrak{B} \subseteq q
  • (b) \forall e \in [\mathfrak{B} ]^{\lt \omega} \exists g_e \in [S]^{\lt \omega} mit \mathfrak{C} = { C_e: e \in [\mathfrak{B}]^{\lt \omega} } cwdick; hierbei C_e = \bigcup_{x \in g_e x^{-1}} D_e.
  • Nenne \mathfrak{B} dann cwpws.

partial Translation

First page

• September 14, 2006
• Continuation: Talk by Sabine Koppelberg
• 3. collectionwise thick (cwt), collectionwise piecewise syndetic (cwpws)
• 3.2 Notation. \mathfrak{B} \subseteq \mathcal{P}(S), \mathfrak{V} := \{ D_e : e \in [\mathcal{B}]^{\lt \omega} \}
  • D_ e = \bigcap_ {B\in e} B
  • In particular, \mathcal{B} \subseteq \mathfrak{V}
• 3.1 Definition
  • For \mathfrak{A} \subseteq \mathcal{P}(S) let Y_ {\mathfrak{A}} := \bigcap_ {A\in\mathfrak{A}} \widehat{A} \subseteq \beta S (closed)
    • [in the setup of 3.2, then Y_ \mathfrak{B} = Y_ \mathfrak{D}]
• 3.3 Theorem & Definition. TFAE:
  • (a) \exists q \in \beta S: \beta S \cdot q \subseteq Y_ \mathfrak{B} (without loss of generality, q\in K(\beta S))
  • (b) \forall D \in \mathfrak{D} D thick.
  • We then call \mathfrak{B} collectionwise thick (cwthick, cwt)
• Proof:
  • =>: \beta S \cdot q \subseteq Y_ {\mathfrak{B}} = Y_ \mathfrak{D} \subseteq \widehat{D} for any D\in \mathfrak{D}
  • <=: For e\in [\mathfrak{B}]^{\lt \omega} take q_e \in \beta S, \beta S \cdot q_e \subseteq \widehat{D_ e}
    • Without loss q\in K(\beta S),
      • (since q_ e \in \beta S \cdot q_ e \in \widehat{D_ e} holds ✓)
    • Let X_ e := \{ q_ f : e \subseteq f \in [ \mathfrak{B} ]^{\lt \omega} \} \subseteq \beta S.

Second page

• Then X_ e \subseteq \widehat{D_ e}
  • [since f\supseteq e \Rightarrow q_ f \in \widehat{D_ f} \subseteq \widehat{D_ e}]
• \{ X_ e : e \in [\mathfrak{B}]^{\lt \omega} \} has the finite intersection property.
• So take q\in \bigcap_ {e \in [ \mathfrak{B} ]^{\lt \omega}} cl_ {\beta S}(X_ e)
• Claim: \forall D \in \mathfrak{D}: \beta S \cdot q \subseteq \widehat{D}
  • [proof]
  • D = D_ e, e\in [ \mathfrak{B}]^{\lt \omega} \Rightarrow X_ e \subseteq \widehat{D_ e}, q\in cl(X_ e) \Rightarrow q \in D_ e
  • \forall S \in S, e \subseteq f, s\cdot q_ f \underset{\beta S \cdot q_ f \subseteq\widehat{D_ f}}{\in} \widehat{D_ f} \subseteq \widehat{D_ e} \Rightarrow s\cdot X_ e \subseteq \widehat{D_ e}
  • \Rightarrow s\cdot q \in \widehat{D_ e} \Rightarrow \beta S \cdot q \subseteq \widehat{D_ e}
  • The claim follows. □
• ["Exercise": construct q as p-limit]
• 3.4 Theorem & Definition. TFAE:
  • (a) \exists q\in K(\beta S) with \mathfrak{B} \subseteq q
  • (b) \forall e \in [\mathfrak{B} ]^{\lt \omega} \exists g_ e \in [S]^{\lt \omega} mit \mathfrak{C} = { C_ e: e \in [\mathfrak{B}]^{\lt \omega} } cwthick; where C_ e = \bigcup_ {x \in g_ e x^{-1}} D_ e.
  • We then call \mathfrak{B} collectionwise piecewise syndetic (cwpws).

Notes

We're back to Sabine Koppelberg's talks about basic \beta S results (with four more pages to come). This time, tackling the not-so-basic notions of collectionwise thick/pws sets. These notions are cricital for analysing sets the minimal ideal -- and equally elusive.

I'm not very happy with notation here; it seems to sacrifice accessibility over corrrectness. A sloppier notation might be helpful. In addition, "collectionwise" is a cumbersome prefix. I'd go for "uniformly" or "coherently" as they are often used in the context of filters (and this is what "collectionwise" is all about). But it probably wouldn't help to add yet another terminology.

Funny thing. I actually spent my last few weeks in Michigan thinking about these notions.