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Transactions of the American Mathematical Society

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



$ H$-closed extensions. II

Authors: Jack R. Porter and Charles Votaw
Journal: Trans. Amer. Math. Soc. 202 (1975), 193-209
MSC: Primary 54D35
MathSciNet review: 0365493
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Abstract: The internal structure and external properties (in terms of other $ H$-closed extensions) of the Fomin extension $ \sigma X$ of a Hausdorff space $ X$ are investigated. The relationship between $ \sigma X$ and the Stone-Čech compactification of the absolute of $ X$ is developed and used to prove that a $ \sigma X$-closed subset of $ \sigma X\backslash X$ is compact and to show the existence of a Tychonoff space $ Y$ such that $ \sigma X\backslash X$ is homeomorphic to $ \beta Y\backslash Y$. The sequential closure of $ X$ in $ \sigma X$ is shown to be $ X$.

It is known that $ \sigma X$ is not necessarily projectively larger than any other strict $ H$-closed extension of $ X$; a necessary and sufficient condition is developed to determine when a $ H$-closed extension of $ X$ is projectively smaller then $ \sigma X$. A theorem by Magill is extended by showing that the sets of $ \theta $-isomorphism classes of $ H$-closed extensions of locally $ H$-closed spaces $ X$ and $ Z$ are lattice isomorphic if and only if $ \sigma X\backslash X$ and $ \sigma Z\backslash Z$ are homeomorphic. Harris has characterized those simple Hausdorff extensions of $ X$ which are subextensions of the Katětov extension. Characterizations of Hausdorff (not necessarily simple) extensions of $ X$ which are subextensions of $ H$-closed extensions $ \theta $-isomorphic and $ S$-equivalent to the Katětov extension are presented.

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Keywords: $ H$-closed extension, Katětov extension, Fomin extension, absolute, projective cover
Article copyright: © Copyright 1975 American Mathematical Society