Boundedness Properties in Function-Lattices

Abstract

The continuous real functions on a topological space X are partially ordered in a natural way by putting ƒ ≦ g if and only if ƒ(x)≦ g(x) for all x in X. With respect to this partial ordering these functions constitute a lattice, the lattice operations ∪ and ∩ being defined by the relations (ƒ ∪ g) (x) = max (ƒ(x), g(x)) ƒ ∩ g )(x) = min (ƒ(x), g(x)). The lattice character of any partially ordered system merely expresses the existence of least upper and greatest lower bounds for any finite set of elements in the system. Many partially ordered systems enjoy much stronger boundedness properties than these: for example, every subset with an upper bound may have a least upper bound, as in the case of the real number system.