# Definition of an arrangement of the plane

... 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).

The right half of the $$x$$-axis is one of the components. It is neither open nor closed as a subset of the plane, but is open as a subset of the axis - that is, it's relatively open.