A measurable cardinal associated with an epireflective subcategory of Hausdorff spaces

Let R be an epireflective subcategory of Hausdorff spaces (i.e., productive and closed-hereditary) containing the 2-point space. = sup { | D | | D  discrete , D ∈ R } , the value ∞ permitted. We show (2.2) if σ ≠ ∞ , then σ is a measurable cardinal (and ∞ and all measurable cardinals arise in this way); (3.1) σ governs the degree to which R is closed under various topological operations. This generalizes known relations between R = α -compact spaces for which σ= the first measurable cardinal ⩾α, and between R = topologically complete spaces for which σ = ∞ , to arbitrary R with its σ.