Let $\mathbb{D}(\mathbb{R}^d)$ be the space of all cadlag functions $\mathbb{R}_{+}\rightarrow\mathbb{R}^d$. Consider the local uniform topology $$ \delta_{lu}(\alpha,\beta)=\sum_{n=1}^{\infty}2^{-n}\,(1\wedge||\alpha-\beta||_n) $$ where $||\alpha||_t=\sup_{0\leq s\leq t}|\alpha(s)|$. In Chapter VI of this book it is said that $(\mathbb{D}(\mathbb{R}^d),\delta_{lu})$ is not separable because the family $\{\alpha_s=1_{[s,\infty)}(t)|s\in[0,1)\}$ is uncountable but $\delta_{lu}(\alpha_s,\alpha_{s^{\prime}})=1/2$ for all $s\neq s^{\prime}$. Can someone explain to me why? I now that separability of a space $X$ means the existence of a countable subset $S$ such that $\overline{S}=X$. Why the aforementioned example exclude the existence of such a countable subset?
Addendum
thanks to a comment I know that the solution comes from this link and this link.
