This title may not explain my question right but I could not think of any better short explanation. My question is, if there is a possibility of a structure (or space) with the dimension $Dim = 3,5$, $Dim = 4,8...$ or any other real numbers value? It's seems pretty obvious that there is no such thing like a $4,5$ dimensional sphere but I'm interested in a proof that no such structure can exist. Does a real dimensional space make any physical sense? Thanks for your help.