Convex semi-lattices of continuous functions

Extract

The purpose of this note is to settlo a question that was left incompletely solved in the author's paper [1].

Let E be a compact Hausdorff space, and C(E) the class of all continuous real-valued functions on E. A subset F of C(E) is called an upper semi-lattice (or, more briefly, is said to admit ں) if ƒ ں g ε F whenever ƒ, g ε F. A subset F of C(E) is said to be upper semi-equicontinuous (u.s.e.c.) if for every ε > 0 and s ε E there exists a neighbourhood U of s such that

We say that F is locally u.s.e.c. if every uniformly bounded subset of F is u.s.e.c.