... for a finite set $H$ of lines in the plane, the arrengement of $H$ is a partition of the plane into relatively open convex subsets, the faces of the arrangement. In this particular case,(*) the faces are the vertices (0-faces), the edges (1-faces), and the cells (2-faces).
[Lecture Notes on Discrete Geometry by Jirka Matousek, page 126].
(*) The "particular" case refers to the $2$ dimensional case.
The question is how come the vertices (0-faces) and edges (1-faces) are relatively open. Indeed if we take some vertex $v$ in the partition, it is not relatively open, is it? (The same for the edges).