弱函数代数与函数代数的插补集

Interpolation Set of Weak Function Algebra and Function Algebra

证明当X为有限离散拓扑空间时,只要A是X上的弱函数代数就有A=C(X);给出了第一可数、紧T_2空间X上的所有弱的数代数的公共插补集;同时就紧T_2空间X上的函数代数A,讨论了与A=C(X)等价的三个命题。

If A is a weak function algebra on a finite discrete topologi-cal space X, thea A=C(X). Let X be a first, countability, compact Hausdorfd space, F be a closed subset of X. Then F is a interpolation set of allweak function algebras on X if and only if F is a finite set. Let A be a function algebra on a compact Hausdorff space X.' Then the followingfour propositions is equivalent 1) A=C(X). 2) For arbitrary x0,y0 ∈X, x0≠y0, there are an f∈A and a straight l such that f(Xl)≠f(y0) and f (x)∈l for x∈X. 3) For arbitrary x0, y0∈eX, x0≠y0 there are an f ∈ee A and a constant d0such that f(x0)≠f(y0), |f(x)|=1 and d0≤arg f(x)≤d0+2π for x∈X.4) For arbitrary x0, y0∈X, x0≠y0 there are a positive integer n and a function f such that f∈A and | f+c| '∈A for arbitrary constant c.