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 f = ∑ i = 1 n g i ∘ ϕ i for some ϕ i ∈ Φ and g i ∈ C ( R ) ( i = 1 , … , n ). Define basic ( X ) = min { | Φ | : Φ is a basic family for X } . If X is separable metrizable then either X is locally compact and finite-dimensional, and basic ( X ) < ℵ 0 , or basic ( X ) = c . s compact and finite-dimensional then basic ( K ) ⩽ cof ( [ w ( K ) ] ℵ 0 , ⊆ ) , and if K contains a discrete subset D with | D | = w ( K ) , then either K is finite-dimensional, and basic ( K ) = cof ( [ w ( K ) ] ℵ 0 , ⊆ ) or basic ( K ) = | C ( K ) | = w ( K ) ℵ 0 .