Sober Scott spaces are not always co-sober. (English) Zbl 1459.06004
The main results of the paper are a generalization of the work done in the papers [D. Zhao and X. Xi, Math. Proc. Camb. Philos. Soc. 164, No. 1, 125–134 (2018; Zbl 1469.06012); D. Zhao, “Poset models of topological spaces”, in: Proceeding of the international conference on quantitative logic and quantification of software. Global-Link Publisher. 229–238 (2009)]. The new main results are presented as Proposition 2.1., Proposition 2.3. and Proposition 2.5. for Scott dcpo models for \(T_{0}\)-spaces and \(d\)-spaces. Then the following result is derived:
Theorem 2.7. Let \(X\) be a \(d\)-space. Then there is a dcpo \(P_{X}\) and a topological embedding \(\psi:X \rightarrow \sum P_{X}\) satisfying the following properties:
Theorem 2.7. Let \(X\) be a \(d\)-space. Then there is a dcpo \(P_{X}\) and a topological embedding \(\psi:X \rightarrow \sum P_{X}\) satisfying the following properties:
- (1)
- \(\psi(X)=\uparrow\psi(X)\);
- (2)
- \(P_{X}=\downarrow\psi(X)\);
- (3)
- \(X\) is sober if and only if \(\sum P_{X}\) is sober.
Reviewer: Seithuti Philemon Moshokoa (Pretoria)
MSC:
| 06B35 | Continuous lattices and posets, applications |
| 06B30 | Topological lattices |
| 54B99 | Basic constructions in general topology |
| 54D30 | Compactness |
Citations:
Zbl 1469.06012References:
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