π-embeddings and Dugundji extension theorems for generalized ordered spaces

✍ Scribed by Yasunao Hattori

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 811 KB
• Volume: 84
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(97)00081-3

✦ Synopsis

We study generalized ordered spaces in which every closed subspace is r-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) = {z E X:

Then every closed subspace of X is n-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, 2) + C(X,Z)

such that for each f E 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).