Red workbook, p14

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Transcript

• Beweis: (a) => (b): \mathfrak{B} \subseteq q \Rightarrow \mathfrak{D} = \{ D_e : e \in [ \mathfrak{B} ]^{\lt \omega} \} \subseteq q
  • Nimm g_e \in [S^{\lt \omega}] mit \beta S \cdot q \subseteq \bigcup_{x \in g_e} \widehat{ x^{-1} D_e} = \widehat{C_e}
  • [ Verfeinerung von pws: A pws \in q \in K(\beta S) \Rightarrow \exists e \in [S^{\lt \omega}]: \beta S \cdot q \subseteq \bigcup x^{-1} A]
  • \Rightarrow \beta S \cdot q \subseteq \bigcap_{C \in \mathfrak{C}} \widehat{C_e} = Y_e \stackrel{3.3}{\Rightarrow} Beh.
• (b) => (a): seien g_e, C_e, \mathfrak{C} wie in (b).
  • Nimm p \in \beta S mit L := \beta S \cdot p \subseteq Y_{\mathfrak{C}} oBdA p\in K(\beta S).
  • \Rightarrow p\in L.
  • Fuer e \in [\mathfrak{B}]^{\lt \omega}: p \in \widehat{C_e} = \bigcup_{x\in g_e} \widehat{x^{-1} D_e}
  • \Rightarrow \exists x_e \in g_e: p \in \widehat{ x_e^{-1}D_e}.
• ((?) wieso eDe?) Nimm w\in \beta S mit S_e := \{ x_f : f \supseteq e, f \in [ \mathfrak{B}]^{\lt \omega} \} \underset{?}{\in} \omega (\forall e \in \beta S)
• Setze q = w \cdot p \Rightarrow q \in \beta S \cdot p \subseteq K(\beta S) und
• es ist \mathfrak{B} \subseteq q [B\in \mathfrak{B} zeige: B\in q = w\cdot p. Aber e:{B} \in [\mathfrak{B}]^{\omega}
  • D_e = B, S_e \in \omega; Fuer alle f\supseteq e: x_f \cdot p \in \widehat{D_f}
  • \widehat{D_f} \subseteq \widehat{D_e} = \widehat{B} \Rightarrow S_e \cdot p \subseteq \widehat{B}
  • \Rightarrow w \cdot p \in \widehat{B}]
• Also folgt die Behauptung.□

partial Translation

• Proof:
• (a) => (b):
  • \mathfrak{B} \subseteq q \Rightarrow \mathfrak{D} = \{ D_e : e \in [ \mathfrak{B} ]^{\lt \omega} \} \subseteq q
  • Take g_e \in [S^{\lt \omega}] with \beta S \cdot q \subseteq \bigcup_{x \in g_e} \widehat{ x^{-1} D_e} = \widehat{C_e}
    • [A pws, A \in q \in K(\beta S) \Rightarrow \exists e \in [S^{\lt \omega}]: \beta S \cdot q \subseteq \bigcup_{x\in e} x^{-1} A]
  • \Rightarrow \beta S \cdot q \subseteq \bigcap_{C \in \mathfrak{C}} \widehat{C_e} = Y_e \stackrel{3.3}{\Rightarrow} the claim.
• (b) => (a): let g_e, C_e, \mathfrak{C} as in (b).
  • Then take p \in \beta S mit L := \beta S \cdot p \subseteq Y_{\mathfrak{C}}; without loss p\in K(\beta S).
  • \Rightarrow p\in L.
  • For e \in [\mathfrak{B}]^{\lt \omega}: p \in \widehat{C_e} = \bigcup_{x\in g_e} \widehat{x^{-1} D_e}
  • \Rightarrow \exists x_e \in g_e: p \in \widehat{ x_e^{-1}D_e}.
  • Take w\in \beta S with S_e := \{ x_f : f \supseteq e, f \in [ \mathfrak{B}]^{\lt \omega} \} \in \omega (\forall e \in \beta S)
  • Now define q = w \cdot p \Rightarrow q \in \beta S \cdot p \subseteq K(\beta S) and
  • since \mathfrak{B} \subseteq q
    • [B\in \mathfrak{B} show: B\in q = w\cdot p. But e:{B} \in [\mathfrak{B}]^{\omega}
    • D_e = B, S_e \in \omega; For all f\supseteq e: x_f \cdot p \in \widehat{D_f}
    • \widehat{D_f} \subseteq \widehat{D_e} = \widehat{B} \Rightarrow S_e \cdot p \subseteq \widehat{B}
    • \Rightarrow w \cdot p \in \widehat{B}]
  • The claim follows.

Notes

This page contains the proof of Theorem 3.4 of the previous part (I guess I should've included that yesterday). I can't really make much of it. It's the dull of writing up a new notion. But if you look closer, you might stumble over a few details (as I did when I took these notes). Writing this up just now I find the choice of w quite striking.