%0 Journal Article
%T Connections between C(X) and C(Y), where Y is a subspace of X
%J Bulletin of the Iranian Mathematical Society
%I Iranian Mathematical Society (IMS)
%Z 1017-060X
%A Aliabad, A.
%A Badie, M.
%D 2011
%\ 12/15/2011
%V 37
%N No. 4
%P 109-126
%! Connections between C(X) and C(Y), where Y is a subspace of X
%K $z$-filter
%K prime
$z$-ideal
%K prime $z^circ$-ideal
%K $P$-space
%K quasi $P$-space
%K $F$-space
%K $CC$-space
%K $G_delta$-point
%R
%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 $fin C(X)$, 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 $Phi : beta Yrightarrowbeta X$ of the identity map $imath: Yrightarrow X$ is one-to-one, for each $z$-embedded subspace $Y$ of $X$. Supposing $p$ is a non-isolated $G_delta$-point in $X$ and $Y=Xsetminus{p}$, 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 $Y$.
%U http://bims.iranjournals.ir/article_374_7ec399b754105013093d1f6f8694836b.pdf