1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Continuous functions in topology

  1. Oct 28, 2007 #1
    1. The problem statement, all variables and given/known data
    In topology, a f: X -> Y is continuous when

    U is open in Y implies that f^{-1}(U) is open in X

    Doesn't that mean that a continuous function must be surjective i.e. it must span all of Y since every point y in Y is in an open set and that open set must have a pre-image open in X, so there must be some x in X s.t. f(x) = y?

    What is wrong with my logic?

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Oct 28, 2007 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    If you think it isn't true, the try and construct a counterexample. :smile:

    If it's true, this might help you see why. If it's false, analysis of the counterexample might help you spot the flaw.
  4. Oct 28, 2007 #3

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    So you're asking about the case when there is no x in X st... That would seem to be the empty set. Now what was the definition of topology?
  5. Oct 28, 2007 #4
    I think matt grime answered it, but my counter example is:

    f(x) = abs(x)

    the set (-4, -2) is open but when when you pull it back you get the null set, which is open in the usual topology on R. I think I see now!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook