Results 1 to 6 of 6

Math Help - Continuity & inverse image of closed sets being closed

  1. #1
    Member Last_Singularity's Avatar
    Joined
    Dec 2008
    Posts
    157

    Continuity & inverse image of closed sets being closed

    Question: Let f be a function defined on a closed domain D. Show that f is continuous if and only if the inverse image of every closed set is a closed set.

    One approach is to solve it using the fact that the complement of open sets are closed. However, I am wondering how I can prove it using the definition of a closed set using limit points?

    I know that a set is closed if it contains all of its limit points. How could I use this definition to prove the above question? In other words, how do I show that if a set in the range holds all of its limit points, mapping it backwards to the domain will create a set also containing all of its limit points?

    Thanks a lot!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Hello,

    Here is something I did : http://www.mathhelpforum.com/math-he...790-post2.html , using neighbourhoods (continuous => ...)
    The key for that is : a set is open if it is a neighbourhood of any of its point.

    I'm not used to limit points... so I hope it will sort of help you !
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,913
    Thanks
    1760
    Awards
    1
    Quote Originally Posted by Last_Singularity View Post
    Question: Let f be a function defined on a closed domain D. Show that f is continuous if and only if the inverse image of every closed set is a closed set.
    I know that a set is closed if it contains all of its limit points.
    What sort of space are you working with? Is a metric space?
    If it is a general top-space, if so what properties do it have?
    To use the limit point approach you need to consider that.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member Last_Singularity's Avatar
    Joined
    Dec 2008
    Posts
    157
    Quote Originally Posted by Plato View Post
    What sort of space are you working with? Is a metric space?
    If it is a general top-space, if so what properties do it have?
    To use the limit point approach you need to consider that.
    This is real analysis, so f: X \rightarrow Y where X,Y \subseteq \mathbb{R}
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,913
    Thanks
    1760
    Awards
    1
    Quote Originally Posted by Last_Singularity View Post
    This is real analysis, so f: X \rightarrow Y where X,Y \subseteq \mathbb{R}
    Point number 1. Real analysis is about more than \mathbb{R} so why would you expect us to know in what setting your problem is casted?

    Now to your question, for continuous functions we have the following.
    The inverse image of an open set is an open set.
    The complement of the inverse image is the inverse of the complement of the image.
    The complement of a closed set is open.
    If M is closed then M^c is open.
    f^{ - 1} \left( {M^c } \right) = \left( {f^{ - 1} (M)} \right)^c
    Can you finish?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member Last_Singularity's Avatar
    Joined
    Dec 2008
    Posts
    157
    Yes, thanks a lot!

    And sorry about not specifying that it was in a real analysis context; I was not aware that such concepts can be generalized to metric spaces and other areas of mathematics...
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. prove that if T is closed then the inverse image is closed
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 24th 2011, 05:48 PM
  2. Metric spaces, open sets, and closed sets
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: March 16th 2011, 06:17 PM
  3. Image of bounded operator is closed
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: June 3rd 2010, 02:03 PM
  4. Metric Space, closed sets in a closed ball
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: November 19th 2009, 06:30 PM
  5. closed sets and continuity
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: August 13th 2009, 04:16 PM

Search Tags


/mathhelpforum @mathhelpforum