1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    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!

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


    User Avatar
    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook