Abstract
We prove that if a countably compact space X X has an open cover U = ⋃ { V n : n > ω } \mathcal {U} = \bigcup {\{ {\mathcal {V}_n}:n > \omega \} } such that each x ∈ X x \in X is in at least one but not more than countably many elements of some V n {\mathcal {V}_n} , then some finite subcollection of U \mathcal {U} covers X X . We apply the theorem in proving several metrization theorems for countably compact spaces and discuss consequences of weak δ θ \delta \theta -refinability, a concept implicit in the statement of the theorem.
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