What is better, 'double connected domain' or 'doubly connected domain'?

Solution 1:

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.

Solution 2:

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.