Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X37No. 420111215Connections between C(X) and C(Y), where Y is a subspace of X109126374ENA.AliabadM.BadieJournal Article20091129In this paper, we introduce a method by which we <br />can find a close connection between the set of prime $z$-ideals <br />of $C(X)$ and the same of $C(Y)$, for some special subset $Y$ of $X$. <br />For instance, if $Y=Coz(f)$ for some $fin C(X)$, then there <br />exists a one-to-one correspondence between the set of prime <br />$z$-ideals of $C(Y)$ and the set of prime $z$-ideals of $C(X)$ <br />not containing $f$. Moreover, considering these relations, we <br />obtain some new characterizations of classical concepts in the <br />context of $C(X)$. For example, $X$ is an $F$-space if and only if <br />the extension $Phi : beta Yrightarrowbeta X$ of the identity <br />map $imath: Yrightarrow X$ is one-to-one, for each $z$-embedded <br />subspace $Y$ of $X$. Supposing $p$ is a non-isolated <br />$G_delta$-point in $X$ and $Y=Xsetminus{p}$, we prove that <br />$M^p(X)$ contains no non-trivial maximal $z$-ideal if and only if <br />$pinbe X$ is a quasi $P$-point if and only if each point of <br />$beta Y setminus Y$ is a $P$-point with respect to $Y$.http://bims.iranjournals.ir/article_374_7ec399b754105013093d1f6f8694836b.pdf