# Tag Archives: b-scale

Nik Weaver asked for a direct proof of the fact that Todorcevic’s axiom implies the failure of CH fails. Here goes. Notation. For a set $X$, we write $[X]^2$ for the set of unordered pairs $\{ \{x,x’\}\mid x,x’\in X, x\neq … Continue reading Posted in Blog, Expository | Tagged , | 13 Comments ## The S-space problem, and the cardinal invariant$\mathfrak b$Recall that an S-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. In a previous post, we showed that such a space exists after adding a Cohen real. Here, we shall construct one from an arithmetic … Continue reading Posted in Blog, Expository | Tagged , | Leave a comment ## c.c.c. vs. the Knaster property After my previous post on Mekler’s characterization of c.c.c. notions of forcing, Sam, Mike and myself discussed the value of it . We noticed that a prevalent verification of the c.c.c. goes like this: given an uncountable set of conditions, … Continue reading Posted in Blog | Tagged , , , | Leave a comment ## Dushnik-Miller for regular cardinals (part 2) In this post, we shall provide a proof of Todorcevic’s theorem, that$\mathfrak b=\omega_1$implies$\omega_1\not\rightarrow(\omega_1,\omega+2)^2\$. This will show that the Erdos-Rado theorem that we discussed in an earlier post, is consistently optimal. Our exposition of Todorcevic’s theorem would be … Continue reading

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## Infinite Combinatorial Topology

Back in 2005, as a master student, I attended a course by Boaz Tsaban, entitled “Infinite Combinatorial Topology”. A friend and I decided to produce lecture notes, but in a somewhat loose sense, that is: we sometimes omit material given … Continue reading