Extending families of disjoint zero sets

Abstract:

The $z$-cellularity $z(X)$ of a space $X$ is defined as \[ z(X) = \sup \left \{ {\left | Z \right |:Z \subset Z(X)} \right \}\] where $Z(X)$ is the family of zero sets of $X$. It is proved using CH that a Tychonoff space $S$ is $T{C^ * }$-embedded in every Tychonoff space it is $C$-embedded in iff $z(S) \leqslant c$. A space $S$ is defined to be $T{C^ * }$-embedded in a space $X$ if any disjoint family of zero sets of $S$ can be extended to a family of disjoint zero sets of $X$. Similar theorems are proved for ${C^ * }$-embedding when $S$ is a $P$-space or the zero sets have the Isiwata property.