Yasunao Hattori

Dimension and superposition of bounded continuous functions on locally compact, separable metric spaces

Abstract Let n be an integer with n ≥ 1 and X be an n -dimensional, locally compact, separable, metric space. Then there are 2 n + 1 continuous real-valued functions φ 1 ,..., φ 2 n +1 of X such that each bounded, continuous, real-valued function ƒ of X is representable in the form ƒ( x ) = Σ 2 n +1 i = 1 g i ( φ i ( x )), x ∈ X , where g i ∈ C ( R ), i = 1,..., 2 n +1. This gives a solution to a problem of Sternfeld.

Lawson topology of the space of formal balls and the hyperbolic topology

Let (X,d) be a metric space and BX=XxR denote the partially ordered set of (generalized) formal balls in X. We investigate the topological structures of BX, in particular the relations between the Lawson topology and the product topology. We show that the Lawson topology coincides with the product topology if (X,d) is a totally bounded metric space, and show examples of spaces for which the two topologies do not coincide in the spaces of their formal balls. Then, we introduce a hyperbolic topology, which is a topology defined on a metric space other than the metric topology. We show that the hyperbolic topology and the metric topology coincide on X if and only if the Lawson topology and the product topology coincide on BX.

π-embeddings and Dugundji extension theorems for generalized ordered spaces

Abstract 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).