[math-fun] Connectedness

Hi all, 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 ? bye Dave (Makin' Magic Fractals)