$H$-closed extensions. II

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.