Spaces with a Finite Family of Basic Functions

TL;DR

This paper characterizes finite-dimensional, locally compact, separable metrizable spaces using a finite family of basic functions, providing a complete solution to longstanding problems in the theory of basic functions.

Contribution

It offers a characterization of certain topological spaces via finite basic functions, solving four open problems and questions in the field.

Findings

Characterization of spaces with finite basic families.

Complete solution to four open problems on basic functions.

Addresses questions posed by Sternfeld, Hattori, and others.

Abstract

A space X is finite dimensional, locally compact and separable metrizable if and only if X has a finite basic family: continuous functions Phi_1,...,Phi_n of X to the reals, R, such that for all continuous f from X to R there are g_1,..., g_n in C(R) satisfying f(x)=g_1(Phi_1(x))+g_2(Phi_2(x))+...+g_n(Phi_n(x)) for all x in X. This give the complete solution to four problems on basic functions posed by Sternfeld, as well as questions posed by Hattori and others.