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Extending families of disjoint zero sets


Author: C. E. Aull
Journal: Proc. Amer. Math. Soc. 89 (1983), 510-514
MSC: Primary 54C50; Secondary 54C45, 54D60, 54G05, 54G10
DOI: https://doi.org/10.1090/S0002-9939-1983-0715876-8
MathSciNet review: 715876
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Abstract: The $ z$-cellularity $ z(X)$ of a space $ X$ is defined as

$\displaystyle z(X) = \sup \left\{ {\left\vert Z \right\vert:Z \subset Z(X)} \right\}$

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.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0715876-8
Article copyright: © Copyright 1983 American Mathematical Society

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