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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Small spaces which ``generate'' large spaces


Author: W. W. Comfort
Journal: Proc. Amer. Math. Soc. 104 (1988), 973-980
MSC: Primary 54C20; Secondary 22A05, 54C25
MathSciNet review: 964881
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Abstract: Let $ \underline S$ denote the class of Tychonoff spaces. For $ X \in \underline S$ and $ \underline C \subseteq \underline S$, we say that $ X$ generates large $ \underline C$-spaces if for every cardinal $ \alpha $ there is $ Y \in \underline S$ such that $ Y \in \underline C, X \subseteq Y$, and every $ Z \in \underline{C}$ with $ X \subseteq Z \subseteq Y$ satisfies $ \vert Z\vert > \alpha $. For classes $ \underline C$ which satisfy certain mild and natural conditions, we show for each $ X \in \underline S$ that $ X$ generates large $ \underline C$-spaces iff there is no weakly free $ \underline C$-space over $ X$--i.e., no space $ Y$ such that $ X \subseteq Y \subseteq C$ and every continuous $ f:X \to Z \in \underline C$ extends to a continuous function $ \bar f:Y \to Z$. Among the classes $ \underline C \subseteq \underline S$ which satisfy these conditions for every $ X \notin \underline C$ are the class of pseudocompact Tychonoff spaces and the class of almost compact (= absolutely $ {C^*}$-embedded) Tychonoff spaces.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0964881-X
PII: S 0002-9939(1988)0964881-X
Keywords: Pseudocompact space, almost compact space
Article copyright: © Copyright 1988 American Mathematical Society