On Scott power spaces

Abstract

In this paper, we mainly discuss some basic properties of Scott power spaces. For a T 0 space X, let K(X) be the poset of all nonempty compact saturated subsets of X endowed with the Smyth order. It is proved that the Scott power space ΣK(X) of a well-filtered space X is still well-filtered, and a T 0 space Y is well-filtered iff ΣK(Y ) is well-filtered and the upper Vietoris topology is coarser than the Scott topology on K(Y ). A sober space is constructed for which its Scott power space is not sober. A few sufficient conditions are given under which a Scott power space is sober. Some other properties, such as local compactness, first-countability, Rudin property and well-filtered determinedness, of Smyth power spaces and Scott power spaces are also investigated.