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mjdominus
Your definition of connectedness
... is AFAICT fine, but can be simplified.
Suppose X' != X. Then there exists some x which is in X' but not in X. Since {X,Y} is a partition of S, x must be in Y. Since Y is a subset of Y', x must be in Y'. So x is in the intersection of X' and Y'. But we have declared that X' and Y' are disjoint, so we have a contradiction. So X' = X. Similarly, Y' = Y.
Hence, a set S is disconnected iff there exist disjoint nonempty open sets X and Y whose union is S. Which is the definition I was taught :-)
BTW, the reason I didn't just submit a patch is because it looked like you were making a pedagogical point with your more complex definition.
Aren't you thinking of the definition of a connected space, whereas mjd is describing the circumstances under which a set of points is connected (as a space) under the subspace topology?
In the latter case, the definition's wrong: X' and Y' don't need to be disjoint, only their intersections with the subspace do. And there's no mention of a larger space into which S is embedded.
That definition is one of the current serious problems with the manuscript. There are some tricky issues there.
A subset S of a space X is connected if the space S is connected in the induced topology. But the induced topology considers intersection of open sets with S. The induced open sets might be disjoin even though the original open sets aren't.
David Radcliffe pointed out the following useful example: Let Z be the integers under the finite-complement topology. Then S = {1, 2} is disconnected. But there is no separation X, Y of S for which the two parts are disjoint.
Yes, I realised this after seeing pozorvlak's last comment. Another example would be something like R + {} with open sets 1) the empty set, and 2) X union {}, where X \subseteq R is open in R. In that example the intersection of two non-empty open sets is always non-empty, and yet it has disconnected subsets (any disconnected subset of R).
But isn't there a straightforward fix? Just remove the requirement that X and Y be disjoint, and ask instead that X be disjoint from Y' and Y from X'.
That does perhaps create the problem that, if you wanted to explain why this funny-looking definition is needed, you would need to relax your “no examples bar R^n” policy…
Here's the thing: you haven't introduced the subspace topology at this point. In fact, I can't find it anywhere in the document. This, IMHO, is a good idea: the subspace topology is one of those definitions that sounds sensible but slightly arbitrary until you know enough category theory to see why it's the Right Thing. So I think you should just talk about entire spaces being connected or disconnected. You could give the definition as it currently is, reinforcing the "separated by disjoint open sets" idea, and leave the simplification as an exercise. I'll submit a patch for consideration...
On second thoughts, "it's the unique topology that makes the inclusion map continuous" doesn't require much sophistication, though it's a perspective that's more likely to occur to someone steeped in the categorical approach.
Robin's suggestion was the definition I learned in school, but I had hoped to cut corners or to fudge it somehow. But as we see, that didn't work.
I can't define connectedness Robin's way until after I've discussed closures. I don't want to introduce the subspace topology (at all) and I can't discuss it in terms of continuous maps until after I discuss continuous maps.
I put connectedness up front because I thought it was an intuitively clear property (unlike, say, compactness) with a straightforward definition. I was wrong about the latter. So perhaps the thing to do is to move connectedness later and do interiors and closures first.