TITLE: Why is $L^{\infty}$ not separable?
QUESTION [54 upvotes]: $l^p (1≤p<{\infty})$ and $L^p (1≤p<∞)$ are separable spaces.

What on earth has changed when the value of $p$ turns from a finite number to ${\infty}$?

Our teacher gave us some hints that there exists an uncountable subset such that the distance of any two elements in it is no less than some $\delta>0$. 
Actually I don't understand the question very well, but I hope I have made the question clear enough.
Thank you in advance.

REPLY [6 votes]: With the topology $\|f-g\|_{l^\infty}=\sup_i |f_i-g_i|$ it is easy to show if C is any countable set, and $(f_{ij})_j$ a sequence of elements of $C$, you can take an element $h\in l^\infty$ so that
$h_i= 0$ if $|f_{ii}|>1$, $h_i=2$ otherwise, hence $\|f_{j}-h\|>1/2$ for all $j$. Then no ball of radius $1/2$ around $h$ can contain an element of $C$.