Peter Krautzberger · on the web

Red workbook, p13

Transcript

First page

• 14. Sept. 2006
• Fortsetzung Vortrag SK
• 3. collectionwise thick (cwt, cwdick), collectionwise pws (cwpws)
• 3.2 Notation.
• (-Suetterlin?),
• Also:
• 3.1 Definition
• Fuer abgeschlossen
• [in 3.2, dann ]
• 3.3 Satz & Def. Aequivalent:
• (a) (natuerlich oBdA )
• (b) dick.
• Dann: heisst cwdick (cwd)
• Beweis
• =>: fuer all
• <=: Fuer nimm
• OBdA, , ( gilt ✓)
• Setze .

Second page

• Damit [ ]
• hat eDE (? check)
• nimm
• Beh.
• [
• ]
• Damit folgt die Behauptung □
• ["Aufgabe": konstruiere durch -limiten?]
• 3.4 Satz & Def. Aequivalent
• (a) mit
• (b) mit cwdick; hierbei .
• Nenne 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.
• In particular,
• 3.1 Definition
• For let (closed)
• [in the setup of 3.2, then ]
• 3.3 Theorem & Definition. TFAE:
• (a) (without loss of generality, )
• (b) thick.
• We then call collectionwise thick (cwthick, cwt)
• Proof:
• =>: for any
• <=: For take
• Without loss ,
• (since holds ✓)
• Let .

Second page

• Then
• [since ]
• has the finite intersection property.
• So take
• Claim:
• [proof]
• The claim follows. □
• ["Exercise": construct as -limit]
• 3.4 Theorem & Definition. TFAE:
• (a) with
• (b) mit cwthick; where .
• We then call collectionwise piecewise syndetic (cwpws).

Notes

We're back to Sabine Koppelberg's talks about basic 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.