TY - JOUR
ID - 374
TI - Connections between C(X) and C(Y), where Y is a subspace of X
JO - Bulletin of the Iranian Mathematical Society
JA - BIMS
LA - en
SN - 1017-060X
AU - Aliabad, A.
AU - Badie, M.
AD -
Y1 - 2011
PY - 2011
VL - 37
IS - No. 4
SP - 109
EP - 126
KW - $z$-filter
KW - prime
$z$-ideal
KW - prime $z^circ$-ideal
KW - $P$-space
KW - quasi $P$-space
KW - $F$-space
KW - $CC$-space
KW - $G_delta$-point
DO -
N2 - 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 $Y$.
UR - http://bims.iranjournals.ir/article_374.html
L1 - http://bims.iranjournals.ir/article_374_7ec399b754105013093d1f6f8694836b.pdf
ER -