H-sober spaces and Hofmann-Mislove type theorems

Abstract

A $T_0$-space is called $ω$-sober if every irreducible closed subset having a countable dense subset is the closure of a unique point. This is precisely $H$-sobriety for the countable irreducible subset system $\mathcal R^ω$. We give two natural Hofmann--Mislove representations in the general $H$-sobriety framework: filters induced by $H$-sets of $X$ characterize $H$-sobriety, whereas filters induced by $H$-sets of the Smyth power space characterize super $H$-sobriety. Their relation is expressed by property $Q$. A minimal closed set argument with a countable selection proves that $\mathcal R^ω$ has property $Q$. Consequently, $X$ is $ω$-sober if and only if its Smyth power space is $ω$-sober, equivalently, every countable irreducible compact-family filter has a compact saturated representation. We also obtain equivalent recovery, compact-intersection and representation conditions for an enlarged class containing the proper Scott-open filters with countable open bases and the filters induced by countable irreducible subsets. A further equivalent formulation uses traces on countable subspaces. The filtered-family specializations recover well-filteredness and $ω$-well-filteredness. Explicit examples show that $ω$-sobriety strictly implies $ω$-well-filteredness, which strictly implies representation for countable open bases alone.