On functionally constant subset of X with respect to a subring A of C(X)

Let Y be a subset of a space X. Then the set of all f 2 C(X) such that f jY be constant (countable) is a subalgebra of C(X), denoted by C1(Y) (Cc(Y)) and Y is constant (countable) with respect to a subring A of C(X) if A C1(Y) (A Cc(Y)). If X is a topological space and Y is a connected subset of X, then Cc(X) C1(Y). In particular, if XnY is countable, then A C1(Y) if and only if A Cc(X). It is shown that for any subspace Y of X, R C1(Y) C(X). Moreover, C1(Y) = R if and only if Y is