Minimal Size of Basic Families

A family $\text{bfam}$ of continuous real-valued functions on a space $X$ is said to be {\sl basic} if every $f\in C(X)$ can be represented $f = \sum_{i=1}^n g_i \circ \phi_i$forsome$\phi_i \in \bfam$and$g_i \in C(\R)$($i=1, …, n$)\mathrm{.}Define$\basic (X) = \min {|\bfam| : \bfam$isabasicfamilyfor$X}$. If $X$ is separable metrizable $X$ then either $X$ is locally compact and finite dimensional, and $\text{basic}(X)<{\mathrm{\aleph }}_0$, or $\basic (X) = \mathfrak{c}$\mathrm{.}If$K$iscompactandeither$w(K)$ (the minimal size of a basis for $K$) has uncountable cofinality or $K$ has a discrete subset $D$ with $\mathrm{\mid }D\mathrm{\mid }=w(K)$ then either $K$ is finite dimensional, and $\basic (K) = \cof ([w(K)]^{\aleph_0}, \subseteq)$,or$\basic (K) = |C(K)|=w(K)^{\aleph_0}$.