Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.17926 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909731728130048 |
|---|---|
| author | Chen, Yu Kou, Hui Lyu, Zhenchao Yang, Weiyu |
| author_facet | Chen, Yu Kou, Hui Lyu, Zhenchao Yang, Weiyu |
| contents | We present several equivalent conditions of the continuity of the supremum function from the square of the Scott space of $C(X)$ to itself under mild assumptions, where $C(X)$ denotes the lattice of closed subsets of a $\mathbf{T_0}$ topological space.
We also show that a $\mathbf{T_0}$ space is quasicontinuous (quasialgebraic) iff the lattice of its closed subsets is a quasicontinuous (quasialgebraic) domain by using $n$-approximation. Furthermore, we provide a necessary condition for when a topological space possesses a Scott completion. This allows us to give more examples which do not have Scott completions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17926 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Scott space of lattice of closed subsets with supremum operator as a topological semilattice Chen, Yu Kou, Hui Lyu, Zhenchao Yang, Weiyu General Topology Logic in Computer Science 54A10, 54A20, 06B35 We present several equivalent conditions of the continuity of the supremum function from the square of the Scott space of $C(X)$ to itself under mild assumptions, where $C(X)$ denotes the lattice of closed subsets of a $\mathbf{T_0}$ topological space. We also show that a $\mathbf{T_0}$ space is quasicontinuous (quasialgebraic) iff the lattice of its closed subsets is a quasicontinuous (quasialgebraic) domain by using $n$-approximation. Furthermore, we provide a necessary condition for when a topological space possesses a Scott completion. This allows us to give more examples which do not have Scott completions. |
| title | The Scott space of lattice of closed subsets with supremum operator as a topological semilattice |
| topic | General Topology Logic in Computer Science 54A10, 54A20, 06B35 |
| url | https://arxiv.org/abs/2503.17926 |