Abstract For infinite cardinals κ λ, let X ϵ C( κ, λ) iff for each open cover U of X of cardinality less than λ, X - U is finally κ-compact (i.e., [κ, β]-compact for all β) for some U ϵ U . The T 3 spaces in C(ω, ω) comprise the NAC spaces of W. Fleissner et al. Their cofinality, cf( X), extends naturally to C(κ, λ) and, combined with a new cardinal invariant, cfc( X), completely determines C(κ, λ), i.e., X ϵ C(κ, λ) iff κ ⩾ cfc( X) and λ ⩽ cf( X). The subclasses C(κ, ω) of cofinally κ-compact and C(κ, κ) of κ-cocompact spaces are more interesting and give meaning to cf( X) and cfc( X). They relate to Dowker space constructions of M.E. Rudin and others, e.g. the inclusion C ( κ, κ) ⊃ C( κ, ω) is made proper by some of these non-ZFC spaces. Section 2 gives equivalents, examples and implications for C(κ, ω), including an extension of a result of P. Nyikos in nonmetric manifolds. Likewise for C(κ, κ) in Section 3. Section 4 develops more fully the relation to [α, β]-compact.