Closed or open set

**Connected** · Apr 4th 2011, 05:24 PM

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.

**Drexel28** · Apr 4th 2011, 05:56 PM

Originally Posted by Connected

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.

**Connected** · Apr 4th 2011, 06:12 PM

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?

**Drexel28** · Apr 4th 2011, 06:30 PM

Originally Posted by Connected

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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