Results 1 to 3 of 3

Math Help - Prove that an extension is a continuous extension

  1. #1
    Member
    Joined
    Oct 2009
    Posts
    195

    Prove that an extension is a continuous extension

    Here is my proof so far. I am lost on how to show that g is continuous!



    thank you so much!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    Re: Prove that an extension is a continuous extension

    Quote Originally Posted by davismj View Post
    Here is my proof so far. I am lost on how to show that g is continuous!



    thank you so much!
    I think you are making this slightly more difficult than it need to be, it's a little hard to follow precisely what's being said. It looks like (and this is the correct approach) that you're defining \widetilde{f}:X\to \mathbb{R} by the rule \widetilde{f}_E=f and by defining f(\lim x_n)=\lim f(x_n) for \lim x_n\in X-E where x_n is some sequence in E. Is this correct? If not, could you explain more.

    I think perhaps most of your questions could be answered by this blog post of mine.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2009
    Posts
    195

    Re: Prove that an extension is a continuous extension

    I'm following the hint given in baby Rudin. I do think its way more difficult, but I followed it because I was having trouble showing that the extension g is actually continuous. But that's exactly the idea. In particular, it uses sequences of sets instead of sequences of numbers. Take a set near x with things in E. You can take those sets as small as you want, and the closure of the image of those sets under f is always non-empty, and in fact there is only on element in every such set.

    Thus, for x in X we associate the unique element y in R such that y is that unique element. It's pretty simple and straightforward, but the problem is showing that g is continuous. Why is it true that any open set around y is the image of an open set in x? Or why is it that for any e there is a d such that d(v,w) < d implies d(g(v),g(w)) < e.

    Your stuff here looks like it answers my question exactly. I'm about to go out but when I get back I'll read through it.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. extension
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 28th 2011, 08:07 AM
  2. Replies: 2
    Last Post: November 23rd 2011, 12:54 AM
  3. Continuous extension at a point
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 23rd 2011, 12:58 AM
  4. Find and prove conductor of cyclotomic extension
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: April 5th 2010, 11:24 PM
  5. Continuous Extension...?
    Posted in the Calculus Forum
    Replies: 4
    Last Post: October 13th 2008, 07:29 PM

Search Tags


/mathhelpforum @mathhelpforum