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Math Help - How to show B[0,1] is not separable

  1. #1
    Junior Member teuthid's Avatar
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    How to show B[0,1] is not separable

    I'm studying for comps, and this one has me stumped:

    Show that the space of bounded functions f:[0,1]\rightarrow \mathbb{R} under the sup norm is not separable.

    I suspect that you'd either have to show an arbitrary dense set is uncountable or that an arbitrary countable set can't be dense, but I don't have any idea how to actually implement those strategies.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: How to show B[0,1] is not separable

    For all x\in [0,1] consider the family of bounded functions f_x(t)=\begin{Bmatrix} 1 & \mbox{ if }& t=x\\0 & \mbox{if}& t\neq x\end{matrix}
    We verify d(f_x,f_y)=1 for all x\neq y i.e., the family \mathcal{F}=\{B(f_x,1/2):x\in[0,1]\} is pairwise disjoint. Now, choose A\subset B[0,1] dense and use that [0,1] is non denumerable.
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