Nowadays, I am just studying the book wrote by Joram Lindenstrauss and Yoav Benyamini,i.e. Geometric Nonlinear Functional Analysis. The putfroward "maximal ε-separated set".I really can not understand this generalization.But I can not find this word in any other books. If you know this definition,please tell me.Thank you.
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$\begingroup$ I'd guess it is a set of vectors $F$ with $\|f-g\|\geqslant \varepsilon$ for all distinct $f,g\in F$ which is maximal subject to this property. $\endgroup$– Tomasz KaniaCommented Oct 28, 2014 at 11:24
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To summarize:
Let $(X,d)$ be a metric space. A subset $\mathcal{A}$ is said to be a maximal $\epsilon$-separated set iff
- For every $x\neq y\in \mathcal{A}$ we have $d(x,y) \geq \epsilon$.
- For every $z\in X\setminus \mathcal{A}$ there exists $w\in \mathcal{A}$ with $d(z,w) < \epsilon$.
The first condition defines $\epsilon$-separation. The second condition defines maximality: it is equivalent to saying that any other $\mathcal{B}$ that is $\epsilon$-separated and such that $\mathcal{B} \supseteq \mathcal{A}$ is in fact $\mathcal{A}$.
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1$\begingroup$ This is one possible meaning, and the one I would have chosen. However, it is conceivable that "maximal" means "of the largest possible size". $\endgroup$ Commented Oct 28, 2014 at 12:49
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$\begingroup$ @Gerald: Is it clear that there is something like that? Isn't it conceivable that there are such sets of size $\aleph_n$ for each $n$, but there is none of size $\aleph_\omega$? $\endgroup$– Asaf Karagila ♦Commented Oct 28, 2014 at 13:35
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1$\begingroup$ Probably that other definition would be used only when it is known that there is a finite upper bound. Compact metric spaces and such places. $\endgroup$ Commented Oct 28, 2014 at 13:40
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1$\begingroup$ @GeraldEdgar: in the context of nonlinear functional analysis, I'd expect we are working over some sort of vector space with characteristic 0, in which case finite upper bound seems unlikely. Besides, wouldn't size give a total ordering (assuming AC or what not) and in which case it makes more sense to say maximum instead of maximal? $\endgroup$ Commented Oct 29, 2014 at 8:52
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$\begingroup$ @GeraldEdgar: I should add also that the only time the phrase "maximal $\epsilon$-separated set" appears in Lindenstrauss and Benyamini is on the first page of chapter 2, in the middle of a proof. They "defined" it using basically the sentence mentioned by Tomek Kania; I inferred the correct definition of maximality from the proof, where they used exactly the second bullet point as one of the steps of the argument. $\endgroup$ Commented Oct 29, 2014 at 8:55