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], x_{1} and x_{2} are in X and A = {f ∈ C(X); f(x_{1}) = f(x_{2})}.