Completeness properties in a broader context

We give a characterization of cofinal local compactness for all totally dis- connected cosmic spaces. As an application of this result we prove that an ultrafilter ξ ∈ βω\ωis a P -point in βω\ω if and only if the space ω ∪ {ξ} is cofinally locally compact. We also show that a subspace X of an ordinal is cofinally locally compact if and only if ext(X ) ≤ ω and every closed countable subspace of X is locally compact. It is established that, for any GO space X , if is Čech-complete (metrizable) for any discrete set D ⊂ X × X , then X is Čech-complete (or metrizable respectively). Besides, it is proved that discrete Čech-completeness of a metric space implies its Baire property. Our results provide an answer to a published open question.