Let $X$ be all real-valued bounded functions on $[0,1]$ with the supremumsmetric.
Prove this:
a) If it has a countable basis $\{B_i\}$ then $ \{ U_{1/2} (x_i ) = \{x \in X | d(x,x_i ) <1/2 \} \} $ (with a fixed $x_i$ in a $B_i$ for each $i$ ) is an open cover of $X$.
b )We define,for each $a$ in $[0,1]$ $f_a (x) := 1$ , if $a=x$ and $f_a (x)=0$ else. Then $d(f_a , f_b) =0$ if $a=b$ and 1 else. Now from this I have to show that $X$ can't have a countable basis.
Please help me.Thanks