I'm just going to hazard a guess, but I suspect that the following discrete subset D of 3-space would require a set of at least (Hausdorff) dimension (dim_H) = 2 to be added to it in order to obtain a connected set. Namely, let discrete set D := {(x,y,z) = (K/N, L/N, 1/N) | (K,L,N) in ZxZxZ+}. As a discrete set, dim_H(D) = 0. Since its limit points constitute R^2 x {0}, and since dim_H(QxQ) = 2 (Q = rationals), I suspect that any countable dense subset of R^2 has Hausdorff dimension = 2, and so intuitively I'd guess that nothing of lower Hausdorff would suffice to add to D so as to form a connected set. --Dan David M. asked: << Myself and another member at http://www.fractalforums.com/ have been having a small discussion about connectedness. Specifically the idea that given any disconnected set of a given dimension d then it's always possible to construct a set with dimension d+1 such that the new set is connected - if considering fractional dimensions then make that floor(d)+1. I have never done any topology beyond the very basics nor any other math relating to general connectedness and am just wondering if the above is correct and if so how goes the proof, when was it proved and who proved it ?
Sometimes the brain has a mind of its own.