Ordinal compactness

We extend to ordinal numbers the more usual compactness notion defined in terms of cardinal numbers. Definition 1. Suppose that X is a nonempty set and that τ is a nonempty family of subsets of X. If α and β are nonzero ordinal numbers, we say that (X, τ) is [β, α]- compact if and only if the following holds. Whenever (O δ ) δ ∈ α is a sequence of members of τ such that S δ ∈ α O δ = X, then there is H ⊆ α with order type < β and such that S δ ∈ H O δ = X. When α and β are both cardinals, X is a topological space, and τ is the topology on X, we get back the classical cardinal compactness notion. See [1] for references. We show that ordinal compactness is a much more varied notion than cardinal com- pactness. We prove a great deal of results of the form “Every [β, α]-compact space is [β0, α0]-compact”, for various ordinals β, α, β0 and α0. Usually, we are able to furnish counterexamples showing that such results are the best possible ones. [1] J. E. Vaughan, Some properties related to [a,b]-compactness, Fundamenta Mathematicae, vol. 87 (1975), pp. 251-260.