From: ram@zedat.fu-berlin.de   
      
   Peter Fairbrother  wrote or quoted:   
   >It is claimed that a topology O on a space M is the simplest structure   
   >which affords a notion of continuity. Two questions:   
   >1] is there a proof of this?   
      
     That might not be the kind of statement you could really prove,   
     since you would need some shared sense of what continuity means   
     "before it has a definition". In math though, a word only gets   
     its meaning once it is pinned down by a definition.   
      
     Continuity gets defined for topological spaces, and that is   
     a very broad definition, since lots of the spaces people care   
     about in practice are also topological spaces.   
      
     Nobody's managed to water down the conditions any further and   
     still come up with a definition of continuity that lines up   
     with what mathematicians expect.   
      
   >2] what other structures on spaces (considered as sets of points) give   
   >notions of continuity?   
      
     Well, I actually know a structure that is not a topological space,   
     but it still has its own idea of "continuity". The elements   
     are not really thought of as "points" anymore. It comes from   
     lattice theory and its connection to denotational semantics.   
      
     To really get that definition, you need a solid book on the topic,   
     but it goes something like: "If D and E are cpo's, the function   
     f is continuous iff it is monotone and preserves lubs of chains,   
     i.e., for all chains d0 c d1 c . . . in D, it is the case that   
     f(U(n>=0): d_n) = U(n>=0): f(d_n) in E.".   
      
     The term was probably picked because of the loose analogy to   
     continuity in topological spaces, and there was no real risk   
     of mixing the two up.   
      
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