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

Document Type : Research Paper



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 $Y$.