Coincidence of the upper Vietoris topology and the Scott topology

Abstract

For a T0 space X, let K(X) be the poset of all non-empty compact saturated sets of X with the reverse inclusion order. The space X is said to have property Q if for any K1,K2∈K(X), K1≪K2 in K(X) iff K2⊆intK1. In this paper, we give several connections among the well-filteredness of X, the sobriety of X, the local compactness of X, the core compactness of X, the property Q of X, the coincidence of the upper Vietoris topology and Scott topology on K(X), and the continuity of x↦↑x:X⟶ΣK(X) (where ΣK(X) is the Scott space of K(X)). It is shown that for a well-filtered space X for which its Smyth power space PS(X) is first-countable, the following three properties are equivalent: the local compactness of X, the core compactness of X and the continuity of K(X). It is also proved that for a first-countable T0 space X in which the set of minimal elements of K is countable for any compact saturated subset K of X, the Smyth power space PS(X) is first-countable. For the Alexandroff double circle Y, which is Hausdorff and first-countable, we show that its Smyth power space PS(Y) is not first-countable.