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Let X be a compact Hausdorff space and A a complex linear algebra of continuous complex-valued functions defined on X. Suppose A is normal on X, i.e., for every pair of disjoint closed sets Ko, Ki in X, there exists a function /G-4 such that /(2£ 0) = 0 and f(Ki) = 1. Does it follow that every continuous complex-valued function on X can be uniformly approximated by functions in A? With the additional assumption that A is closed under complex conjugation, it follows by the Stone-Weierstrass theorem. (Trivially, if A is normal then A separates points.) The same theorem implies that the analogous question in the case of real-valued functions has an affirmative answer. However, in the complex-valued case it need not be so. An example will be given which demonstrates this. In this example, the space X is a suitably chosen compact set in the complex plane. The algebra is R(X), the algebra of all functions which can be uniformly approximated on X by rational functions whose poles lie outside X. It will be shown that R(X) is normal on X and is a proper sub-algebra of C(X) t the algebra of all continuous complex-valued functions on X. Since R(X) is closed under uniform limits, this will be sufficient. Two lemmas are needed to accomplish this. One is a modification of an observation of Mergelyan [l]. The second represents a slight extension of a result due to Beurling [2].

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