Red workbook, p13
25 Mar 2014
Source
Red Workbook, p.13, part 1
Red Workbook, p.13, part 2
Transcript
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}]^{<\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}]^{<\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} ]^{< \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}]^{< \omega} }
hat eDE (? check)
nimm
q\in \bigcap_{e \in [ \mathfrak{B} ]^{<\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}]^{< \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} ]^{<\omega} \exists g_e \in [S]^{< \omega}
mit
\mathfrak{C} = { C_e: e \in [\mathfrak{B}]^{<\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}]^{<\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}]^{<\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} ]^{< \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}]^{< \omega} }
has the finite intersection property.
So take
q\in \bigcap_ {e \in [ \mathfrak{B} ]^{<\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}]^{< \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} ]^{<\omega} \exists g_ e \in [S]^{< \omega}
mit
\mathfrak{C} = { C_ e: e \in [\mathfrak{B}]^{<\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.