π-embeddings and Dugundji extension theorems for generalized ordered spaces

We study generalized ordered spaces in which every closed subspace is π-embedded and which satisfy the Dugundji Extension Theorem. We prove: Let X be a perfectly normal generalized ordered space in which the set E(X) = {x ϵ X: (→,x] or [x,→) is open in X} is σ-discrete in X. Then every closed subspace of X is π-embedded. Furthermore, for every closed subspace A of X and for any locally convex linear topological space Z there is a linear transformation u : C(A,Z) → C(X,Z) such that for each f ϵ C(A,Z), u(f) is an extension of f and the range of u(f) is contained in the closed convex hull of the range of f. This is a partial answer to a question asked by Heath and Lutzer (1974).