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

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

How do I do it? I need a push.