A measurable cardinal associated with an epireflective subcategory of Hausdorff spaces

Let σ=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.