... 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).


Imagine the plane arrangement determined by the two coordinate axes. The four open quadrants are open sets in the plane. The other five sets in the partition are the halves of each axis and the origin.

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.

The origin is in fact an open subset of the set consisting only of the origin, so relatively open.


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