Re: Connectedness question

Ryan Reich wrote: 

some deep topology

I think a course in topology is not required to transfer what is on stake here. Both in Q and R, there is a concept of distance: the distance between numbers a and b is |a - b|, the absolute difference of both numbers. More generally, other sets can have a concept of distance:
the distance between a and b is denoted as d(a, b) usually. A set is complete if the following holds: if a_1, a_2, a_3, ... is a Cauchy sequence (google for definition), then it converges to a limit within the set. Now 1/0!, 1/0!+1/1!, 1/0!+1/1!+1/2!, 1/0!+1/1!+1/2!, ... is a Cauchy sequence that converges to e = exp(1). But e is not a rational number. So Q is not complete. R is complete, and in fact, R is the smallest complete set that contains Q, and therefore called the completion of Q.
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