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Scott topology on Smyth power posets.
- Source :
- Mathematical Structures in Computer Science; Oct2023, Vol. 33 Issue 9, p832-867, 36p
- Publication Year :
- 2023
-
Abstract
- For a $T_0$ space X , let $\mathsf{K}(X)$ be the poset of all nonempty compact saturated subsets of X endowed with the Smyth order $\sqsubseteq$. $(\mathsf{K}(X), \sqsubseteq)$ (shortly $\mathsf{K}(X)$) is called the Smyth power poset of X. In this paper, we mainly discuss some basic properties of the Scott topology on Smyth power posets. It is proved that for a well-filtered space X , its Smyth power poset $\mathsf{K}(X)$ with the Scott topology is still well-filtered, and a $T_0$ space Y is well-filtered iff the Smyth power poset $\mathsf{K}(Y)$ with the Scott topology is well-filtered and the upper Vietoris topology is coarser than the Scott topology on $\mathsf{K}(Y)$. A sober space Z is constructed for which the Smyth power poset $\mathsf{K}(Z)$ with the Scott topology is not sober. A few sufficient conditions are given for a $T_0$ space X under which its Smyth power poset $\mathsf{K}(X)$ with the Scott topology is sober. Some other properties, such as local compactness, first-countability, Rudin property and well-filtered determinedness, of Smyth power spaces, and the Scott topology on Smyth power posets, are also investigated. [ABSTRACT FROM AUTHOR]
- Subjects :
- PARTIALLY ordered sets
TOPOLOGY
COMMERCIAL space ventures
Subjects
Details
- Language :
- English
- ISSN :
- 09601295
- Volume :
- 33
- Issue :
- 9
- Database :
- Complementary Index
- Journal :
- Mathematical Structures in Computer Science
- Publication Type :
- Academic Journal
- Accession number :
- 173587267
- Full Text :
- https://doi.org/10.1017/S0960129523000257