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Continous Mappings (Functions)

  1. Feb 15, 2006 #1

    JasonRox

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    I have a question regarding this.

    I wish I were home right now so I can give the exact words.

    Anyways, the book is talking about continuous maps from one space to another. This is basically what it says...

    Let f be a function with domain D in R. Then the following statements are equivalent:
    f is continuous
    If D is open, then the inverse image of every open set under f is again open.
    If D is closed, then the inverse image of every closed set under f is again closed.

    There are others, but that's not important.

    I just want to clarify that f does not need to be one-to-one, correct?
     
  2. jcsd
  3. Feb 15, 2006 #2

    matt grime

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    No, f does not need to be one to one. Instead of 'inverse image' try using the phrase 'preimage' of a set as someone here once pointed out. Can't remember who to credit it with.
     
  4. Feb 15, 2006 #3

    JasonRox

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    So, using the "preimage" we can "map" to two elements?

    I used quotes for map because it doesn't really satisfy the definition.
     
    Last edited: Feb 15, 2006
  5. Feb 15, 2006 #4

    HallsofIvy

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    I'm not sure what you mean by "map" two elements but here is an example:

    If f(x)= x2 (not one-to-one) and D is the interval (-1, 4) then f-1(D) = (-2, 2), the set of numbers whose square is between -1 and 4. Since (-1, 4) is open and f is continuous, that inverse image (preimage if you prefer) is open.
     
  6. Feb 15, 2006 #5

    JasonRox

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    That's exactly what I wanted to confirm. Thanks.
     
  7. Feb 16, 2006 #6

    HallsofIvy

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    My most embarassing moment: My first semester in graduate school, I was called on to do a proof in Topology class that involved f-1(A) where A is a set. Without thinking, I did the proof assuming that f had an inverse! (Hey, it said f-1!)
     
  8. Feb 16, 2006 #7

    JasonRox

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    I'll remember this whenever I get embarrased. :wink:

    Were you lucky enough to redo your proof?
     
  9. Feb 16, 2006 #8

    HallsofIvy

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    I did later submit a proof on paper to the professor but that day I just shrank into a small lump in my chair. The next time I was called on to present a proof in class I did well and I actually passed the course!
     
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