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Homework Help: Funtion continuity and open sets

  1. Apr 11, 2010 #1
    1. The problem statement, all variables and given/known data

    Suppose that [tex]f : (X,d_X) \to (Y,d_Y)[/tex]. If [tex]f[/tex] is continuous,
    must it map open sets to open sets? If [tex]f[/tex] does map open sets to
    open sets must [tex]f[/tex] be continuous?

    2. Relevant equations



    3. The attempt at a solution

    The answer to the first question is yes. The answer to the second question I guess is "no". Is this correct? How can I prove it?
     
  2. jcsd
  3. Apr 11, 2010 #2

    HallsofIvy

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    Science Advisor

    Neither is true. For example, a constant function is trivially continuous but maps any set, including any open set, to a singleton set which is not, generally, open.

    Conversely, if B has the discrete topology, so that all sets are open, and A is a [0, 1] with the usual topology, the function f(x)= a for all [itex]0\le x< 1/2[/itex] and f(x)= b for all [itex]1/2\le 0\le 1[/itex], where a and b are distinct points in B, is not continuous but maps open sets to open sets.
     
    Last edited by a moderator: Apr 11, 2010
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