Presumably you require some relationship between the new set and the given disconnected set. (Otherwise, for example, a unit d+1-dimensional cube would work.) What is that relationship? On 9/2/2011 3:35 PM, David Makin wrote:

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)

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Hi, To be honest I lack the formal education for "correct" technical description - I think I mentioned previously that my formal pure maths education ended at Further Maths "A" level some 30 years ago and I haven't really studied further myself beyond what helps in creating aesthetically pleasing images of fractals (or better algorithms for doing so) ;) One can obviously draw a line to join any number of objects but said line would then be within the original dimensions of the set (unless the original were just a set of points in which case you've upped the integer dimension by 1 anyway) - so it would be a different set when considered in that dimensionality, but if the line were offset into an extra dimension the objects could be connected in this new set of higher dimension such that the original is still unchanged as a lower -dimensional cut/slice of the new set. I hope that's clear enough ;) bye Dave On 2 Sep 2011, at 20:35, David Makin wrote:

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)

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