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'Doubly connected domain' returns more in google search, but it looks more strange for my ear. Which is the correct form according to rules so far?

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This is somewhat technical mathematically, but I've tried to keep it simple - I hope the mathematicians will overlook the imprecision, and non-mathematicians will bear with the detail...

A simply-connected domain is a shape in which, if you take any two points, there will be just one path taking you from one to the other (where you consider two paths being as the same if one can be smoothly deformed into the other).

A doubly-connected domain is a shape in which there are two paths between any two points, and those paths are distinct and can't be smoothly deformed into each other.

So this latter domain is, in some sense, "twice as connected" as the former, and so it is called doubly-connected.

Calling it a "double connected domain" would instead imply that you had two simply-connected domains, or a simply-connected domain that was in some way doubled, which is not the sense that is desired.

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So (if we’re allowed to get technical in the comments) what’s the actual definition? Is it that π₁(X) ≅ ℤ, or that π₁(X) ≅ ℤ/(2), or something else? I’ve heard the term in passing plenty of times, but never come across the precise definition, and quick googling doesn’t find one… –  PLL Mar 10 '11 at 14:31
    
@PLL: I have to confess it's been a long time since I was 100% up to speed on this. I've only ever seen the term used referring to ℂ, and all of the definitions I can find refer to "a connected shape with a single hole". If I remember my topology rightly (and I'm prepared to be corrected :) then this (or at least, a more precise version of it) is equivalent to my definition (or at least, a more precise version of it, with suitable conditions - such as non-self-intersection of the paths - added). If my memory is failing me please do point out any errors! –  psmears Mar 10 '11 at 15:23
    
The ambiguity that I wasn’t sure about is that if you have two paths [up to homotopy], you usually have many more. E.g. in the circle, there are two obvious paths from -1 to 1: go around the top, or around the bottom. But you could also go “seventeen times around clockwise, then over the top”, and so on: there are in fact ℤ-many paths, π₁(X) ≅ H₁(X) ≅ ℤ, and this is what “connected with a single hole” would usually be read as, I think. It you ask for literally exactly two paths, i.e. π₁(X) ≅ ℤ/(2), then this would instead capture only somewhat more subtle spaces [cont’d] –  PLL Mar 10 '11 at 16:03
    
like the projective plane. So if (as you suggest) shapes like the circle or annulus are what the definition is meant to capture, then the intuition “essentially two different paths” has to be amended slightly, via the idea “essentially one obstacle between paths”, into the formal definition “ℤ-many paths”, i.e. either “π₁(X) ≅ ℤ” or “H₁(X) ≅ ℤ”. –  PLL Mar 10 '11 at 16:07
    
@PLL: Yes, that is what I was (trying to) get at with the "non-self-intersecting" condition... with the circle, if you restrict the paths to non-self-intersecting ones, then you get exactly two paths ("go left" and "go right"). I was thinking that this condition could be extended for the annulus case, but having slept on it I'm much less convinced! However I believe the definition can be made to work for annuli if you restrict the paths to be non-intersecting loops (i.e. start=end), and fiddle with the conditions a little more (eg considering clockwise/anticlockwise as the same)... –  psmears Mar 12 '11 at 14:50
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Doubly connected domain (or doubly-connected, with a hyphen) is the right phrase.

Two general tips, here.  Firstly, when using Google as a test for something like this, search with the whole phrase in quotes, i.e. search for "doubly connected domain", not just doubly connected domain.  With the quotes, google will search for the phrase as a whole — in this, it correctly shows doubly connected domain as much more common than double.  Without the quotes, it just searches for pages in which the individual words all appear, not necessarily together — so it could give the wrong answer here, just because in most contexts, double is more common than doubly.

Secondly, this is a technical term.  For technical terms, use a technical reference — check a good textbook, or a couple of academic papers, that talk about these domains, and see what they use.  It could be that, for instance, in mathematics people talk about doubly-connected, but that engineers have a completely different thing called double-connected.  Then, Google wouldn’t know which one you’re after, and nor would I.  Using a technical reference from your field makes sure that you’ve got the right term for the things you’re talking about.

Edit: Oh, and a third point, a grammatical one.  Doubly here is modying connected — it’s telling you what kind of connectedness property the domain satisfies.  Since it’s modifying an adjective, it has to be an adverb — i.e. doubly is the right form, not double.  Contrast a phrase like compact connected domain, in which both compact and connected are modifying the noun domain, so are both adjectives.  This point is subsidiary to my second one, though — sometimes technical terms bend the usual conventions of grammar a bit, and in those cases, the technical usage is the one to follow.

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I think your third point is the most important point in your answer. "Double connected domain" means something totally different than "Doubly connected domain" because of the grammar, and thus the "right" answer depends on the context. –  Mr. Shiny and New 安宇 Mar 10 '11 at 16:59
    
Thanks. Nice to see how this applies to "simply connected domain" and "doubly connected domain" comparing to "single layer potential" and "double layer potential". –  V_V Mar 12 '11 at 12:18
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Doubly connected domain is the right answer.

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