Hofmann-Mislove type definitions of non-Hausdorff spaces
2022
One of the most important results in domain theory is the Hofmann-Mislove Theorem, which reveals a very distinct characterization for the sober spaces via open filters. In this paper, we extend this result to the d-spaces and well-filtered spaces. We do this by introducing the notions of Hofmann-Mislove-system (HM-system for short) and
$\Psi$
-well-filtered space, which provide a new unified approach to sober spaces, well-filtered spaces, and d-spaces. In addition, a characterization for
$\Psi$
-well-filtered spaces is provided via
$\Psi$
-sets. We also discuss the relationship between
$\Psi$
-well-filtered spaces and H-sober spaces considered by Xu. We show that the category of complete
$\Psi$
-well-filtered spaces is a full reflective subcategory of the category of
$T_0$
spaces with continuous mappings. For each HM-system
$\Psi$
that has a designated property, we show that a
$T_0$
space X is
$\Psi$
-well-filtered if and only if its Smyth power space
$P_s(X)$
is
$\Psi$
-well-filtered.
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