In [The sup = max problem for the extent of generalized metric spaces, Comment. Math. Univ. Carolin. The special issue devoted to Čech 54 (2013), no. 2, 245–257], the author and Yajima discussed the sup = max problem for the extent and the Lindelöf degree of generalized metric spaces: (strict) -spaces, (strong) -spaces and semi-stratifiable spaces. In this paper, the sup = max problem for the Lindelöf degree of spaces having -diagonals and for the extent of spaces having point-countable bases...
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.
It looks not useful to study the sup = max problem for extent, because there are simple examples refuting the condition. On the other hand, the sup = max problem for Lindelöf degree does not occur at a glance, because Lindelöf degree is usually defined by not supremum but minimum. Nevertheless, in this paper, we discuss the sup = max problem for the extent of generalized metric spaces by combining the sup = max problem for the Lindelöf degree of these spaces.
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