Minimal size of basic families

A family Φ of continuous real-valued functions on a space X is said to be basic if every f∈C(X) can be represented for some ϕi∈Φ and gi∈C(R) (i=1,…,n). Define . If X is separable metrizable then either X is locally compact and finite-dimensional, and basic(X)<ℵ0, or basic(X)=c.If K is compact and finite-dimensional then basic(K)⩽cof(ℵ0[w(K)],⊆), and if K contains a discrete subset D with |D|=w(K), then either K is finite-dimensional, and basic(K)=cof(ℵ0[w(K)],⊆) or basic(K)=|C(K)|=wℵ0(K).