부산대학교 도서관

The Hofmann-Mislove (HM) Theorem gives a distinct characterization ofsober spaces via the behavior of open filters of open sets (in thelattice of open sets). In this paper the authors vastly expand thisapproach to consider generalizations of three basic types of$T_0$-spaces: sober spaces, well-filtered spaces, and $d$-spaces. Theydo this by introducing what they call HM-systems $\Psi$ of sets of openfilters of the lattice of open sets, which are sandwiched between theminimal filters, which have as elements the neighborhoods of compactsaturated sets, and the maximal set of all open filters. TheseHM-systems are used to define and study $\Psi$-well-filtered spaces andto give characterizations of sober and $d$-spaces from the latter. Theauthors further show that the category of complete $\Psi$-well-filteredspaces is a full reflective subcategory of the category of$T_0$-spaces. They close with the result that a $T_0$-space is$\Psi$-well-filtered if and only if its Smyth power space is$\Psi$-well-filtered.