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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Topological extension properties

Author: R. Grant Woods
Journal: Trans. Amer. Math. Soc. 210 (1975), 365-385
MSC: Primary 54D35
MathSciNet review: 0375238
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Abstract: It is known that if a topological property $ \mathcal{P}$ of Tychonoff spaces is closed-hereditary, productive, and possessed by all compact $ \mathcal{P}$-regular spaces, then each $ \mathcal{P}$-regular space $ X$ is a dense subspace of a space $ {\gamma _\mathcal{P}}X$ with $ \mathcal{P}$ such that if $ Y$ has $ \mathcal{P}$ and $ f:X \to Y$ is continuous, then $ f$ extends continuously to $ {f^\gamma }:{\gamma _\mathcal{P}}X \to Y$. Such topological properties are called extension properties; $ {\gamma _\mathcal{P}}X$ is called the maximal $ \mathcal{P}$-extension of $ X$. In this paper we study the relationships between pairs of extension properties and their maximal extensions. A basic tool is the concept of $ \mathcal{P}$-pseudocompactness, which is studied in detail (a $ \mathcal{P}$-regular space $ X$ is $ \mathcal{P}$-pseudocompact if $ {\gamma _\mathcal{P}}X$ is compact). A classification of extension properties is attempted, and several means of constructing extension properties are studied. A number of examples are considered in detail.

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Keywords: Topological extension property, maximal $ \mathcal{P}$-extension, $ \mathcal{P}$-regular, $ \mathcal{P}$-pseudocompact, $ \mathcal{D}$-compact, $ {\aleph _0}$-bounded
Article copyright: © Copyright 1975 American Mathematical Society