Tag: ccc

Using Dushnik-Miller to prove that every sigma-compact group is ccc

(This is the write-up for a talk I gave in the Toronto Student Set Theory and Topology seminar on May 2, 2013.)

A couple of weeks ago I gave a talk for the set-theoretic topology course I was in, on the topic of cardinal invariants of topological groups. While I was preparing that presentation I discovered the following fact:

Theorem [Tkachenko, 1983] Every \sigma -compact group is ccc.

I will present a proof that I have adapted from Tkachenko’s original paper (“Souslin property in free topological groups on bicompacta”) and the proof that appears in Arhangel’skii & Tkachenko’s big purple book (Section 5.3 of Topological Groups and Related Structures). Both proofs involve first proving a Ramsey result about covers of a space, then using this to prove that a particular space has “weak-precalibre \aleph_1 ” (i.e. Property K) which is a property that implies ccc. Learning this proof has been part of my ongoing attempt to learn how Ramsey results show up in topology.

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Stevo’s Forcing Class Fall 2012 – Class 11

(This is the eleventh lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the tenth lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. As always I appreciate any type of feedback, including reporting typos, in the comments below.)

Summary

  • Baumgartner’s Axiom is introduced.
  • A combinatorial lemma is proved, which really shows that a certain poset is proper.
  • Some words are said about the productivity of chain conditions.

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Stevo’s Forcing Class Fall 2012 – Class 10

(This is the tenth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the ninth lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. As always I appreciate any type of feedback, including reporting typos, in the comments below. I am sure that this one has a ton of typos. There are also some omissions which I will try to patch up.)

Summary

  • We define the Side Condition Method.
  • We force the P-Ideal Dichotomy.
  • A cardinal preserving modification of the side condition is discussed.
  • Semi-properness is introduced, and then used to prove a fact about projectively stationary sets.
  • The Strong Reflection Principle is introduced.
  • Some consequences are discussed.
  • A ZFC version of diamond is stated.

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Stevo’s Forcing Class Fall 2012 – Class 9

(This is the ninth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the eighth lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. As always I appreciate any type of feedback, including reporting typos, in the comments below.)

Summary

  • We define the notion of properness.
  • Show that the poset \in -Collapse is proper.
  • Justin Moore gives some intuition.
  • Explain the Open Colouring Axiom.
  • Give a related proper poset.

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Stevo’s Forcing Class Fall 2012 – Class 8

(This is the eighth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the seventh lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. I didn’t take these notes, so there are few Stevo quotes. Thanks to Dana Bartasova for letting me reproduce her notes here. As always I appreciate any type of feedback, including reporting typos, in the comments below.)

Summary

  • Show Chang’s Conjecture implies \theta_2 = \aleph_2 .
  • Show the failure of CC is equivalent to a nice colouring existing. This is fairly technical.
  • Introduce definable posets and “Definable CH”.

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Stevo’s Forcing Class Fall 2012 – Class 7

(This is the seventh lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the sixth lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. As always I appreciate any type of feedback, including reporting typos, in the comments below.)

Summary

  • Talk about the CH “hierarchy”.
  • Point out the MA analogue.
  • Show that it is possible to force a function to be continuous on a large set.
  • Show that Chang’s Conjecture is preserved by ccc forcing.

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Stevo’s Forcing Class Fall 2012 – Class 6

(This is the sixth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the fifth lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. As always I appreciate any type of feedback, including reporting typos, in the comments below.)

Summary

  • We talk about some caliber properties.
  • Establish that the ccc functor really goes to ccc spaces.
  • Use this to create an interesting ccc poset.
  • Talk about some combinatorial versions of CH.
  • Mention a few consequences of CH^3 .

Continue reading Stevo’s Forcing Class Fall 2012 – Class 6

Stevo’s Forcing Class Fall 2012 – Class 1

(In the fall of 2012 I will be taking Stevo Todorcevic’s class in Forcing at the University of Toronto. I will try to publish my notes here, although that won’t always be possible.)

Summary of class 1.

  • Discuss examples of ccc posets whose product is not ccc.
  • Prove a theorem of Baumgartner’s that relates the branches in a tree to its antichain structure.
  • Display the differences in chain conditions.

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