Coincidence of the upper Vietoris topology and the Scott topology

• The conditions are investigated under which the upper Vietoris topology coincides with Scott on K ( X ) . For a well-filtered space having first-countable Smyth power space, and coincide. T 0 in set of minimal elements any compact saturated subset, its is first-countable. Alexandroff double circle, first-countable, not , let be poset all non-empty sets reverse inclusion order. said to have property Q if for 1 2 ∈ ≪ iff ⊆ int In this paper, we give several connections among well-filteredness sobriety local compactness core coincidence continuity x ↦ ↑ : ⟶ Σ (where ). It shown that P S following three properties equivalent: also proved countable subset circle Y Hausdorff show