Sets and Binary Relations

Abstract

In the first sections of this chapter, we provide well-known fundamentals of linear spaces and topological spaces, binary relations and cones. We add some new results and details that are necessary for the understanding of the following chapters and for the proofs therein. Furthermore, our notation is introduced. We investigate properties of recession cones and norms which are defined by means of cones. Statements about the algebraic interior and the algebraic closedness of sets are summarized. We introduce the directional closedness of sets in linear spaces together with related notions and prove direction-depending properties of sets. These properties build the basis for a full characterization of translation invariant functions. They allow to deduce results in cases where usually one of the following conditions has to be fulfilled, but is not satisfied or too restrictive:

- the space is equipped with a topology,

- a set is closed or has a nonempty interior in a given topology,

- a set is algebraically closed or its algebraic interior is nonempty.