Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Continuous, Onto Function: (0,1)x(0,1)->R^2

  1. Aug 6, 2011 #1
    Continuous, Onto Function: (0,1)x(0,1)-->R^2

    Hi, All:

    Just curious about finding a continuous onto function from the open unit square (0,1)x(0,1)
    into R^2. All I can think is that the function must go to infinity towards the edges, because
    if it could be continued into the whole square, so that we would get a continuous bijection between compact [0,1]x[0,1] and Hausdorff R^2, which would then necessarily be a homeomorphism, which cannot happen for many reasons (e.g., square has boundary, R^2 does not; square has a cutset of two points and R^2 does not).

    It seems at first that pairing up two bijections f,g:(0,1)-->R^2 into h=fxg would work, but it does not, I don't think; the technique does not work in the general case, e.g., f(x)=x and g(x)=2x, both from R-->R.

    Any Ideas?

    Thanks.
     
  2. jcsd
  3. Aug 7, 2011 #2
    Re: Continuous, Onto Function: (0,1)x(0,1)-->R^2

    [URL]http://latex.codecogs.com/gif.latex?f(z)=%5Cleft%20(%5Ctan%5Cleft%20%5Cpi%20(x-%5Cfrac{1}{2})%20%5Cright%20%5Cright%20)%20+%5Cmathbf{i}%5Cleft%20(%5Ctan%5Cleft%20%5Cpi%20(y-%5Cfrac{1}{2})%20%5Cright%20%5Cright%20)[/URL]
    where x=Re{z} and y=Im{z}.
    Does it work?
     
    Last edited by a moderator: Apr 26, 2017
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Continuous, Onto Function: (0,1)x(0,1)->R^2
  1. What is: (1/x^0)*x (Replies: 1)

  2. Is 1/x ~ 0? (Replies: 8)

Loading...