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• CommentRowNumber1.
• CommentAuthorTobyBartels
• CommentTimeOct 14th 2012

I separated Cauchy filter from Cauchy space.

• CommentRowNumber2.
• CommentAuthorTobyBartels
• CommentTimeOct 16th 2012

Now with nonstandard analysis.

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeOct 16th 2012

I’ve never encountered the word “adequality” before, and I thought I’d read a fair amount of NSA. I’m looking forward to seeing the page created…

• CommentRowNumber4.
• CommentAuthorTobyBartels
• CommentTimeOct 16th 2012
• (edited Oct 16th 2012)

I could have sworn that we already had it, which is initially why I linked to it. Since the internal search no longer works, I can't easily check whether the string adequal appears anywhere in the Lab, but Google has neither word ‘adequal’ nor ‘adequality’ indexed, and I can't imagine how else that string would appear.

There is a Wikipedia page, but that's a historical treatment focussing on Fermat (who invented the term), not really what I remember seeing before. Google is no help. But I know that I was reading about this (in the context of NSA) somewhere.

Anyway, it simply means that two points are infinitely close together. For example, in a metric space, $x \approx y$ iff $d(x,y)$ is infinitesimal. If $x$ is standard, then $x \approx y$ iff $y$ belongs to the monad of $x$, which makes sense in any topological space (and in fact defines the topology); the relation between arbitrary hyperpoints makes sense in any uniform space (and in fact defines the uniformity). When both $x$ and $y$ are standard, then adequality reduces to the specialisation order (or its symmetrisation; I'm not sure how the term should be used in non-symmetric spaces).

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeOct 16th 2012

Do you pronounce ’adequal’ as “ad+equal”, with approximately equal emphasis on the first and second syllables? (Up until now, it never occurred to me that there was an etymological connection between ’adequate’ and ’equate’, but a quick check in one of my dictionaries seems to indicate that.)

• CommentRowNumber6.
• CommentAuthorTobyBartels
• CommentTimeOct 16th 2012

I did so pronounce it, but now that you mention ‘adequate’, maybe I won't. I've only seen it written.

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeOct 17th 2012

Got it, thanks! I’m familiar with the concept, but I don’t think I’ve heard a name for it before, aside from “infinite closeness”.

• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeOct 19th 2012

Re #5, the Latin ancestor of adequate appears in Aquinas’s

Veritas est adaequatio intellectus et rei.

There’s a lot packed into that ’adequation’ of mind and thing.

• CommentRowNumber9.
• CommentAuthorTobyBartels
• CommentTimeOct 19th 2012

Truth is the adequation of mind and thing, is that what it says?

• CommentRowNumber10.
• CommentAuthorDavid_Corfield
• CommentTimeOct 21st 2012

That’s right.