Results about the standard Skorohod topology on the space $D([0,\infty))$ of cadlag functions from $[0,\infty)$ to $\mathbb{R}^d$, can easily be found in many classic texts, such as Billingsley's Convergence of Probability Measures. Completeness, separability, ... are well explained.
However, I am really struggling to find a reference for the Skorohod topology on the space $D(\mathbb{R})$ of cadlag functions from $\mathbb{R}$ to $\mathbb{R}^d$. Does anybody know of any reference?
The only explicit treatment I could find is in a diploma thesis:
https://www.ruhr-uni-bochum.de/imperia/md/images/mathematik/folder/lehrstuhl-xii/vogel_diplomarbeit_2005.pdf
From what I understand the construction of the Skorokhod metric on $D(\mathbb{R})$ is very similar to the one on $D([0,\infty))$.