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Math Help - Closed or open set

  1. #1
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    Closed or open set

    Consider X=\mathcal C [0,T] with the metric d_\infty(f,g)=\{|f(t)-g(t)|:0\le t\le T\}. Find if \Omega=\{f\in X:|f(t)|\le M\} is closed or open.

    I must check that \forall x_0\in A,\exists r_{x_0} such that B(x_0,r_{x_0})\subset X, in order to be open.

    How do I do it? I need a push.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Connected View Post
    Consider X=\mathcal C [0,T] with the metric d_\infty(f,g)=\{|f(t)-g(t)|:0\le t\le T\}. Find if \Omega=\{f\in X:|f(t)|\le M\} is closed or open.

    I must check that \forall x_0\in A,\exists r_{x_0} such that B(x_0,r_{x_0})\subset X, in order to be open.

    How do I do it? I need a push.
    Here is what I would do. Fix x\in [0,T] and consider the evaluation functional \varphi_x:C[0,T]\to\mathbb{R}:f\mapsto f(x) this is continuous and consequently \varphi_x^{-1}\left([-M,M]\right)=\left\{f\in C[0,T]:-M\leqslant f(x)\leqslant M\right\} (being the preimage of a closed set under a continuous map) is closed. Note though that \displaystyle \Omega=\bigcap_{x\in[0,T]}\varphi_x^{-1}\left([-M,M]\right) so \Omega is closed.
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  3. #3
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    Okay, that's somewhat advanced I had in mind. I think I'll be learning that stuff you exposed according to my class professor.

    Now, is it another way to do it?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Connected View Post
    Okay, that's somewhat advanced I had in mind. I think I'll be learning that stuff you exposed according to my class professor.

    Now, is it another way to do it?
    Come on man...throw me a bone. You've posted like 25 questions in the last two days and I've helped with most of them. Can you, for the love of God, show some work or effort?
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