Connections between C(X) and C(Y), where Y is a subspace of X

In this paper, we introduce a method by which we can find a close connection between the set of prime z-ideals of C(X) and the same of C(Y), for some special subset Y of X. For instance, if Y=Coz(f) for some f in C(X), then there exists a one-to-one correspondence between the set of prime z-ideals of C(Y) and the set of prime z-ideals of C(X) not containing f. Moreover, considering these relations, we obtain some new characterizations of classical concepts in the context of C(X). For example, X is an F-space if and only if the extension Phi : beta Y rightarrow beta X of the identity map i math: Yrightarrow X is one-to-one, for each z-embedded subspace Y of X. Supposing p is a non-isolated G_delta-point in X and Y=X setminus{p}, we prove that M^p(X) contains no non-trivial maximal z-ideal if and only if pinbe X is a quasi P-point if and only if each point of beta Y setminus Y is a P-point with respect to Y