Collectionwise Hausdorffness at limit cardinals
The shrinking property of products of cardinals
The shrinking property and the B-property in ordered spaces
Subparacompactness in locally nice spaces
Countable compactness of lexicographic products of GO-spaces
We characterize the countable compactness of lexicographic products of GO-spaces. Applying this characterization about lexicographic products, we see:
Separating by -sets in finite powers of ω₁
It is known that all subspaces of ω₁² have the property that every pair of disjoint closed sets can be separated by disjoint -sets (see [4]). It has been conjectured that all subspaces of ω₁ⁿ also have this property for each n < ω. We exhibit a subspace of ⟨α,β,γ⟩ ∈ ω₁³: α ≤ β ≤ γ which does not have this property, thus disproving the conjecture. On the other hand, we prove that all subspaces of ⟨α,β,γ⟩ ∈ ω₁³: α < β < γ have this property.
Sequential completeness of subspaces of products of two cardinals
Let be a cardinal number with the usual order topology. We prove that all subspaces of are weakly sequentially complete and, as a corollary, all subspaces of are sequentially complete. Moreover we show that a subspace of need not be sequentially complete, but note that is sequentially complete whenever and are subspaces of .
Normal subspaces in products of two ordinals
Let λ be an ordinal number. It is shown that normality, collectionwise normality and shrinking are equivalent for all subspaces of .
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