A compact Hausdorff space A compact Hausdorff space X and subalgebras of C(X) satisfying in only three conditions of four following conditions:
(a) uniformly closed,
(b) separating the points of X,
(c) containing constant functions,
(d) closed under complex conjugation.

(a), (b), (c); i.e. a uniform algebra:
Consider a compact subset X of C and suppose that A is the uniform closure of rational functions with poles out of X.
(a), (b), (d):
With X = [a,b], let A be the set of all polynomials in one variable, but without constant terms.
(b), (c), (d):
With X = [a,b], put A to be the algebra of all polynomials in one variable.
(a), (c), (d): Let X = [a,b] , x1 and x2 are in X and A = {f Î C(X) ; f(x1) = f(x2)}.

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On 22 Feb 2001, 00:13.