From: peter@tsto.co.uk   
      
   On 26/08/2025 11:00, Stefan Ram wrote:   
   > William Hyde  wrote or quoted:   
   >> Stefan Ram wrote:   
   >>> William Hyde  wrote or quoted:   
   >>>> "A continuous function is one whose inverse maps open sets to open sets"   
   >>> It might be better to ask that the preimages of open sets come   
   >>> out open, because if you think about the sine function, you   
   >>> can still use that test on it even though it does not have an   
   >>> inverse, since that wouldn't be single-valued.   
   >> We're getting well beyond the limits of my memory here.   
   >   
   >    Oh, that's not really a big deal. The whole point is just that   
   >    every function in math, whether it's topology or whatever else,   
   >    has to be set up so each x only matches with one y.   
   >   
   >    Figure 1 shows that with the sine function.   
   >   
   >       |      .+++.   
   >       |     .+   ^.   
   >     y |<---.+     ^.   
   >       |    +|      +   
   >       |   | |       +   
   >       |  .' |        .   
   >       |  |  |        +   
   >       | .'  |         -   
   >       | +   |         |   
   >       |.    |          -   
   >       |+    |          |   
   >       |'    |           -   
   > +++++|++++++++++++++++++   
   >             x   
   >   
   >    But the other way around, a lot of functions end up giving you a   
   >    bunch of possible values, so you no longer have a proper "inverse   
   >    function".   
   >   
   >    Figure 2 shows that again with the sine function.   
   >   
   >       |      .+++.   
   >       |     .+   ^.   
   >     y |----.+-----^.   
   >       |    +|     |+   
   >       |   | |     | +   
   >       |  .' |     |  .   
   >       |  |  |     |  +   
   >       | .'  |     |   -   
   >       | +   |     |   |   
   >       |.    |     |    -   
   >       |+    |     |    |   
   >       |'    V     V     -   
   > +++++|++++++++++++++++++   
   >            x_0   x_1   
   >   
   >    You can still talk about points like "x_0" and "x_1" in the   
   >    figure, but they're just "preimages", not actual outputs of   
   >    an "inverse function".   
      
   Hmmm, as we are talking about sets here, rather than individual points   
   or values, does it matter whether a point in the target has more than   
   one inverse?   
      
   Peter F   
      
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