From: peter@tsto.co.uk   
      
   On 24/08/2025 10:10, Stefan Ram wrote:   
   > 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.   
      
      
   So, simplest known structure, rather than simplest possible structure.   
   Thanks, that's about what I expected.   
      
   >   
   >> 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.   
      
      
   Yikes! I know a bit about lattices as groups (I am a cryptologist), but   
   not as abstract algebra. Though I can read that, but not grok it. I will   
   maybe get a book ..   
      
   Ta again   
      
   Peter Fairbrother   
      
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