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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeFeb 20th 2013
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   Author: TobyBartels
   Format: MarkdownItexI made a stub [[uniform convergence space]], but I need to read the reference.

   I made a stub uniform convergence space, but I need to read the reference.

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeFeb 22nd 2013
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   Author: TobyBartels
   Format: MarkdownItexWe now have motivation (made up by me) but not yet a definition.

   We now have motivation (made up by me) but not yet a definition.

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeFeb 22nd 2013
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   Author: TobyBartels
   Format: MarkdownItexRelated edits to [[Cauchy space]], [[Cauchy filter]], and [[ultrafilter]].

   Related edits to Cauchy space, Cauchy filter, and ultrafilter.

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeFeb 22nd 2013
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   Author: TobyBartels
   Format: MarkdownItexNow with a definition at [[uniform convergence space]].

   Now with a definition at uniform convergence space.

5. • CommentRowNumber5.
   • CommentAuthorTobyBartels
   • CommentTimeFeb 22nd 2013
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   Author: TobyBartels
   Format: MarkdownItexA mistake in the nonstandard stuff at [[Cauchy filter]]. Fixed.

   A mistake in the nonstandard stuff at Cauchy filter. Fixed.

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeFeb 24th 2013
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   Author: TobyBartels
   Format: MarkdownItexNow probably everything that I\'ll write at [[uniform convergence space]] is pretty much there.

   Now probably everything that I'll write at uniform convergence space is pretty much there.

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 26th 2013
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   Author: Mike Shulman
   Format: MarkdownItexNice! The idea of a "filter of filters" takes some wrapping my head around, though.

   Nice! The idea of a “filter of filters” takes some wrapping my head around, though.

8. • CommentRowNumber8.
   • CommentAuthorTobyBartels
   • CommentTimeFeb 26th 2013
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   Author: TobyBartels
   Format: MarkdownItexYeah, I have to keep reminding myself that this isn\'t really more abstract than a Cauchy space, or go back to the example of a metric space, where a uniform filter may be assumed to be (a superset of the eventuality filter of the pairing of) a pair of sequences.

   Yeah, I have to keep reminding myself that this isn't really more abstract than a Cauchy space, or go back to the example of a metric space, where a uniform filter may be assumed to be (a superset of the eventuality filter of the pairing of) a pair of sequences.

9. • CommentRowNumber9.
   • CommentAuthorTobyBartels
   • CommentTimeMay 26th 2013
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   Author: TobyBartels
   Format: MarkdownItexI changed the ad-hoc term "uniform filter" to "asymptotic filter", although this is wrong. There must be *some* term for a pair of sequences $a, b$ such that $\lim_n (a_n - b_n) = 0$, but what is it? I know "coterminal", but this assumes that they converge; and "asymptotic" really means that $\lim_n a_n/b_n = 1$ (at least for positive-valued sequences), which is weaker.

   I changed the ad-hoc term “uniform filter” to “asymptotic filter”, although this is wrong. There must be some term for a pair of sequences $a, b$ such that $\lim_n (a_n - b_n) = 0$, but what is it? I know “coterminal”, but this assumes that they converge; and “asymptotic” really means that $\lim_n a_n/b_n = 1$ (at least for positive-valued sequences), which is weaker.

10. • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 26th 2013
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    Author: DavidRoberts
    Format: MarkdownItexAdditively asymptotic?

    Additively asymptotic?

11. • CommentRowNumber11.
    • CommentAuthorTobyBartels
    • CommentTimeMay 26th 2013
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    Author: TobyBartels
    Format: MarkdownItexI thought of ‘exponentially asymptotic’, since $\exp a \sim \exp b$. But why doesn\'t it have its own term? It\'s a very natural notion.

    I thought of ‘exponentially asymptotic’, since $\exp a \sim \exp b$. But why doesn't it have its own term? It's a very natural notion.

12. • CommentRowNumber12.
    • CommentAuthorTobyBartels
    • CommentTimeJul 2nd 2013
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    Author: TobyBartels
    Format: MarkdownItexBased on the answers (or lack thereof) at [my MathOverflow question](http://mathoverflow.net/questions/135093/terminology-for-sequences-functions-that-approach-each-other), I'm going to stick with ‘asymptotic’, but I added to the terminological warning.

    Based on the answers (or lack thereof) at my MathOverflow question, I’m going to stick with ‘asymptotic’, but I added to the terminological warning.

13. • CommentRowNumber13.
    • CommentAuthorTim_Porter
    • CommentTimeJul 3rd 2013
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    Author: Tim_Porter
    Format: MarkdownItex@Toby: as support for that terminology look at [[asymptotic C-star-homomorphism]]. This notion, due to Connes and Higson, corresponds in the (non-commutative) C*-algebra setting to strong shape morphisms in the corresponding topological one.

    @Toby: as support for that terminology look at asymptotic C-star-homomorphism. This notion, due to Connes and Higson, corresponds in the (non-commutative) C*-algebra setting to strong shape morphisms in the corresponding topological one.