Minimal Size of Basic Families

TL;DR

This paper investigates the minimal size of basic families of continuous functions on topological spaces, revealing that for separable metrizable spaces the size is either finite or continuum, and characterizes this size for certain compact spaces.

Contribution

It provides a complete characterization of the minimal size of basic families for separable metrizable and certain compact spaces, linking it to topological invariants.

Findings

For separable metrizable spaces, the minimal size is finite or continuum.

For certain compact spaces, the minimal size depends on the weight and structure.

The paper establishes explicit formulas for the minimal size in these cases.

Abstract

A family $\bfam$ of continuous real-valued functions on a space $X$ is said to be {\sl basic} if every $f \in C(X)$ can be represented $f = \sum_{i=1}^n g_i \circ \phi_i$ for some $\phi_i \in \bfam$ and $g_i \in C(\R)$ ($i=1, ..., n$). Define $\basic (X) = \min \{|\bfam| : \bfam$ is a basic family for $X\}$. If $X$ is separable metrizable $X$ then either $X$ is locally compact and finite dimensional, and $\basic (X) < \aleph_0$, or $\basic (X) = \mathfrak{c}$. If $K$ is compact and either $w(K)$ (the minimal size of a basis for $K$) has uncountable cofinality or $K$ has a discrete subset $D$ with $|D|=w(K)$ then either $K$ is finite dimensional, and $\basic (K) = \cof ([w(K)]^{\aleph_0}, \subseteq)$, or $\basic (K) = |C(K)|=w(K)^{\aleph_0}$.