부산대학교 도서관

The Hofmann-Mislove (HM) Theorem gives a distinct characterization of sober spaces via the behavior of open filters of open sets (in the lattice of open sets). In this paper the authors vastly expand this approach to consider generalizations of three basic types of $T_0$-spaces: sober spaces, well-filtered spaces, and $d$-spaces. They do this by introducing what they call HM-systems $\Psi$ of sets of open filters of the lattice of open sets, which are sandwiched between the minimal filters, which have as elements the neighborhoods of compact saturated sets, and the maximal set of all open filters. These HM-systems are used to define and study $\Psi$-well-filtered spaces and to give characterizations of sober and $d$-spaces from the latter. The authors further show that the category of complete $\Psi$-well-filtered spaces 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.