ar X iv : 0 90 9 . 45 63 v 1 [ m at h . G N ] 2 4 Se p 20 09 Minimal Size of Basic Families ∗
نویسنده
چکیده
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 = P n i=1 gi ◦ φi for some φi ∈ Φ and gi ∈ C(R) (i = 1, . . . , n). Define basic (X) = min{|Φ| : Φ is a basic family for X}. IfX is separable metrizable X then eitherX is locally compact and finite dimensional, and basic (X) < א0, or basic (X) = 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)]0 ,⊆), or basic (K) = |C(K)| = w(K)0 .
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