1* Introduction* The classical alternation theorem states that G is an n dimensional subspace of C[α, δ], then for each f eC[a, b] and its unique best approximation g0 e G, the error f — g0 has n + 1 alternating peak points. It is natural to ask whether such a result remains valid we replace C[a, b] by C(T), where is an arbitrary compact subset of the real line R or, more generally, by CO(JΓ), where is any locally compact subset of R. [Here C0(T) denotes the Banach space of all real-valued contionuous functions / on vanishing at infinity (i.e., {t e T \f(t) ^ e} is compact for each e > 0), and endowed with the supremum norm: ||/|| = sup ίeτΊ/(ί)|. When is actually compact, we often write C(T) for C0(Γ).] And such a result is not valid, characterize those n dimensional subspaces G of GQ(T) for which the result does hold. Properties (A-l) and (A-2) above, in the special case = [α, 6], have been considered by Jones and Karlovitz [6] who proved that an n dimensional subspace G of C[a, b] has property (A-l) and only G has property (A-2) and only G is weak Chebyshev (i.e. G has property (W-4) defined below). Furthermore, Handscomb, Mayers, and Powell [5; Theorem 8] showed that an n dimensional subspace G of C[a, b] has property (A-3) (if and) only G is a subspace. (The if part is just the classical alternation theorem.) In this paper, for each i e {1, 2, 3}, we give intrinsic characterizations of theose subspaces G of C0(T) which have property (A-i). It turns out that, contrary to the case when — [a, b], properties (A-l) and (A-2) are not the same in general; and property (A-3) does