On Scott power spaces [article]

Xiaoquan Xu, Xinpeng Wen, Xiaoyong Xi
2022 arXiv   pre-print
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
more » ... ober. 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.
arXiv:2207.08720v1 fatcat:g7yeb5qn6fbclj6ug3j5zyhrhe