ABSTRACT
Connections between convexity and (semi-)lattice theory also exist in a different direction. Lattices and semilattices themselves can be convex- ified in several ways, and yield a rich class of examples in convexity to which general results apply. In this chapter, the authors describe how a convex structure induces a family of base-point orders on the underlying set. After discussing some conditional completeness concepts, the people construct an intrinsic topology. They discuss some properties of the intrinsic topology for a class of Helly number 2 convexities, which form a natural generalization of distributive lattices. Finally, the authors deal with continuous selections. Two applications of a general selection theorem of are quoted: one concerning the approximation of lsc functions by continuous functions, and one concerning a lattice-theoretic formulation of the classical Urysohn and Tietze extension theorems in topology.
