The Space $D[0,1]$ of cadlag-functions (right continuous, left side limit exists) from $[0,1]$ to $\mathbb{R}$ with the Sup-norm is not separable.

Question

Attempt: I need to show, that there exists an uncountable amount of open sets, which are disjoint. Then it wouldn't exist a countable subset of $D[0,1]$, which is dense. How do I show that?

Answer 1

For $t\in [0,1]$ let $f_t={1}_{[t,1]}$. If $s\neq t$, then $\lVert f_s-f_t\rVert_\infty=1$. Thus $B_{1/2}(f_s)\cap B_{1/2}(f_t)=\emptyset$ for $s\neq t$ (here $B_r(f)$ denotes the open ball with radius $r$ around $f$), that is, $\{B_{1/2}(f_t)\mid t\in [0,1]\}$ is the uncountable set of open subsets of $D([0,1])$ you were looking for.