Timeline for Extracting a common convergent indexing from an uncountable family of sequences

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21 events

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Dec 13, 2023 at 11:46	comment	added	Pietro Majer		@AndreasBlass Thank you for sharing this Handbook! Talking about Banach spaces, a nice consequence of Prop 8.1 is that the dual space of $\ell_\infty/c_0$ is not w* separable, which immediately implies that $c_0$ is not a factor subspace of $\ell_\infty$ (this is Whitely’s proof)
Dec 12, 2023 at 20:48	comment	added	Joel David Hamkins		@AndreasBlass Great! I suppose it's not surprising that this was already known.
Dec 12, 2023 at 20:43	comment	added	Andreas Blass		$\mathbb c_{\mathbb R}=\mathbb c_{\{0,1\}}=\mathfrak s$ is Theorem 3.2 in my chapter of the Handbook of Set Theory (available at dept.math.lsa.umich.edu/~ablass/hbk.pdf).
Dec 12, 2023 at 0:01	vote	accept	Isaac
Dec 11, 2023 at 13:53	comment	added	Joel David Hamkins		@FarmerS I think that is right, and indeed, it seems $𝕔=𝕔_{\mathbb{R}}=𝕔_{\{0,1\}}=\frak{s}$.
Dec 11, 2023 at 13:42	comment	added	Farmer S		Isn't $c_{\mathbb{R}}=c_{\{0,1\}}$? (See comment below 3rd answer below...)
Dec 10, 2023 at 22:47	comment	added	Joel David Hamkins		I think the followup question about the nature of 𝕔 and even $𝕔_{\mathbb{R}}$ is really interesting, and if we don't get more responses here on this thread, I may ask a separate question about it in a day or two. (If that is all right with Isaac..)
Dec 10, 2023 at 22:44	comment	added	Joel David Hamkins		Yes, my perspective is that the question is really more about: what kinds of subsequence indexing functions are there? This is a purely set-theoretic issue, deeply connected with the combinatorics of the cardinal characteristics of the continuum. I wouldn't be surprised if the space $\mathbb{R}$ will carry the whole phenomenon, so that the core of the question is really about $\mathbb{R}$. But I admit that this could be wrong, since I don't really have strong intuitions or even knowledge about reflexive separable Banach spaces in the weak topology.
Dec 10, 2023 at 22:40	comment	added	alvoi		It's interesting to me that your answer didn't require basically anything about the question: you are not really using the fact that $X$ is a reflexive separable Banach space with the weak topology, but just that every sequence we are considering has a convergent subsequence. I guess you just need some particular notion of convergence. Maybe this can help to focus on the core of the question. I guess that it has something to do with how many "incompatible" convergent subsequences some sequence in the space can have (for $\{0,1\}$ it's 2)...
Dec 10, 2023 at 20:49	comment	added	Joel David Hamkins		Is $𝕔=𝕔_{\mathbb{R}}$?
Dec 10, 2023 at 20:35	history	edited	Joel David Hamkins	CC BY-SA 4.0	added 26 characters in body
Dec 10, 2023 at 20:29	history	edited	Joel David Hamkins	CC BY-SA 4.0	added 26 characters in body
Dec 10, 2023 at 20:20	history	edited	Joel David Hamkins	CC BY-SA 4.0	added 2202 characters in body
Dec 10, 2023 at 20:11	history	rollback	Joel David Hamkins		Rollback to Revision 5
Dec 10, 2023 at 20:04	history	edited	Joel David Hamkins	CC BY-SA 4.0	added 1182 characters in body
Dec 10, 2023 at 19:57	history	edited	Joel David Hamkins	CC BY-SA 4.0	added 1182 characters in body
Dec 10, 2023 at 18:31	history	edited	Joel David Hamkins	CC BY-SA 4.0	added 68 characters in body
Dec 10, 2023 at 17:54	history	edited	Joel David Hamkins	CC BY-SA 4.0	added 132 characters in body
Dec 10, 2023 at 17:45	history	edited	Joel David Hamkins	CC BY-SA 4.0	added 60 characters in body
Dec 10, 2023 at 17:38	history	edited	Joel David Hamkins	CC BY-SA 4.0	added 186 characters in body
Dec 10, 2023 at 17:32	history	answered	Joel David Hamkins	CC BY-SA 4.0

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