2020/03/14 by Xu, Xiaoquan, Yang, Zhongqiang
#FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.2003.06542
For a T0 space X, let \mk (X) be the poset of all compact saturated sets of X with the reverse inclusion order. The space X is said to have property Q if for any K1, K2∈ \mk (X), K2≪ K1 in \mk (X) iff K2⊆ \ii~ K1. 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 \mk (X), and the continuity of x↦\ua x : X \longrightarrow Σ~ \mk (X) (where Σ~ \mk (X) is the Scott space of \mk (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 \mk (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.