\(n\)-H-closed spaces (Q1710636)

A Hausdorff space that is closed in every Hausdorff space in which it is embedded is called H-closed. The theory of H-closed spaces is already well developed. A basic reference is the book of \textit{J. R. Porter} and \textit{R. G. Woods} [Extensions and absolutes of Hausdorff spaces. New York etc.: Springer-Verlag (1988; Zbl 0652.54016)]. One question in this area is finding H-closed extensions of (non-H-closed) Hausdorff spaces. The most prominent of these extensions are the Katětov extension and the Fomin extension. In this paper the theory is generalized to the case of \(n\)-Hausdorff spaces and the corresponding \(n\)-H-closed spaces. A space \(X\) is called \(n\)-Hausdorff if for each set of distinct points \(x_1,x_2,\ldots ,x_n\in X\) there exist open sets \(U_1,U_2,\ldots ,U_n\) such that \(x_i\in U_i\) for each \(i\) and \(\bigcap_{i = 1}^n {U_i }=\emptyset\). It is shown that a space \(X\) is \(n\)-Hausdorff if and only if for every open ultrafilter on \(X\) the set of adherence points has cardinality \(\leq n-1\) . An \(n\)-Hausdorff space \(X\) is called \(n\)-H-closed if \(X\) is closed in every \(n\)-Hausdorff space \(Y\) in which \(X\) is embedded. The authors show that an \(n\)-Hausdorff space is \(n\)-H-closed if and only if for every open ultrafilter on \(X\) the set of adherence points has cardinality \(n-1\) . The main part of the study consists of constructing the maximal \(n\)-H-closed extension as well as the related Fomin extension and to address general questions related to \(n\)-H-closed spaces.