Let A be a uniformly closed point separating algebra of bounded real valued functions on a set X, containing the constant functions. A is called z-separating if whenever Z 1 , Z 2 {Z_1},{Z_2} are disjoint zero sets of members of A there is some f ∈ A f \in A with f ( Z 1 ) = 0 f({Z_1}) = 0 and f ( Z 2 ) = 1 f({Z_2}) = 1 . We prove that A is z-separating if and only if A consists of precisely those bounded real valued functions f on X for which f − 1 ( C ) {f^{ - 1}}(C) is a zero set of some member of A for every closed set C of real line.