H-sober spaces and Hofmann-Mislove type theorems

A T₀-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 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 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.