Results 1 to 3 of 3

Thread: Continuous Functions and Topologies

  1. #1
    Super Member Aryth's Avatar
    Joined
    Feb 2007
    From
    USA
    Posts
    666
    Thanks
    2
    Awards
    1

    Continuous Functions and Topologies

    We have a definition for continuity in terms of open sets of topologies, but I had a question about it. Here's the definition:

    Let $\displaystyle (X,\Omega)$ and $\displaystyle (Y,\Theta)$ be topological spaces. A function $\displaystyle f: X \to Y$ is continuous relative to $\displaystyle \Omega$ and $\displaystyle \Theta$ provided $\displaystyle f^{-1}(U) \in \Omega$ for every $\displaystyle U \in \Theta$.

    My question is this: Is $\displaystyle f^{-1} : Y \to X$ continuous relative to $\displaystyle \Theta$ and $\displaystyle \Omega$ provided $\displaystyle U \in \Theta$ for every $\displaystyle f^{-1}(U) \in \Omega$? Or is there something else you have to do to that sentence to make it true?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,782
    Thanks
    2824
    Awards
    1

    Re: Continuous Functions and Topologies

    Quote Originally Posted by Aryth View Post
    We have a definition for continuity in terms of open sets of topologies, but I had a question about it. Here's the definition:
    Let $\displaystyle (X,\Omega)$ and $\displaystyle (Y,\Theta)$ be topological spaces. A function $\displaystyle f: X \to Y$ is continuous relative to $\displaystyle \Omega$ and $\displaystyle \Theta$ provided $\displaystyle f^{-1}(U) \in \Omega$ for every $\displaystyle U \in \Theta$.

    My question is this: Is $\displaystyle f^{-1} : Y \to X$ continuous relative to $\displaystyle \Theta$ and $\displaystyle \Omega$ provided $\displaystyle U \in \Theta$ for every $\displaystyle f^{-1}(U) \in \Omega$? Or is there something else you have to do to that sentence to make it true?
    This is a case in which the function notation is getting into the way of understanding.

    You have a function $\displaystyle f:X\to Y$.
    But then $\displaystyle f^{-1}$ maps $\displaystyle {P}(Y)\to {P}(X)$, i.e. between power sets.

    Continuity is defined on topological spaces. What is the topology on $\displaystyle {P}(Y)~?$
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,781
    Thanks
    3030

    Re: Continuous Functions and Topologies

    The very first time I was asked to present a proof in a Graduate school class, it was precisely a problem involving $\displaystyle f^{-1}(A)$ where A was in the image of f. I did the whole problem assuming that f was invertible (they said "$\displaystyle f^{-1}$" didn't they?)!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. continuous functions between Topologies
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: Oct 19th 2011, 07:59 AM
  2. [SOLVED] functions of topologies
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Sep 27th 2011, 04:50 PM
  3. Continuous Functions
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Jan 23rd 2010, 06:06 AM
  4. Continuous Functions?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Nov 18th 2009, 01:36 PM
  5. Continuous functions
    Posted in the Calculus Forum
    Replies: 6
    Last Post: Jan 28th 2008, 10:57 PM

Search Tags


/mathhelpforum @mathhelpforum