From: wthyde1953@gmail.com   
      
   Peter Fairbrother wrote:   
   > 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?   
   >   
   > 2] what other structures on spaces (considered as sets of points) give   
   > notions of continuity?   
      
   Centuries ago in real analysis I ran across the theorem that:   
      
   "A continuous function is one whose inverse maps open sets to open sets"   
      
   which is a special case, I think, of the Heine-Borel theorem, which in   
   its more  general form may be relevant to your question.   
      
   William Hyde   
      
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