|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'504'928
Articles rated: 2609
26 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
A complete Heyting algebra whose Scott space is non-sober | | Xiaoquan Xu
; Xiaoyong Xi
; Dongsheng Zhao
; | | Date: |
2 Mar 2019 | | Abstract: | We prove that (1) for any complete lattice $L$, the set $mathcal{D}(L)$ of
all nonempty saturated compact subsets of the Scott space of $L$ is a complete
Heyting algebra (with the reverse inclusion order); and (2) if the Scott space
of a complete lattice $L$ is non-sober, then the Scott space of
$mathcal{D}(L)$ is non-sober. Using these results and the Isbell’s example of
a non-sober complete lattice, we deduce that there is a complete Heyting
algebra whose Scott space is non-sober, thus give a positive answer to a
problem posed by Jung. We will also prove that a $T_0$ space is well-filtered
iff its upper space (the set $mathcal{D}(X)$ of all nonempty saturated compact
subsets of $X$ equipped with the upper Vietoris topology) is well-filtered,
which answers another open problem. | | Source: | arXiv, 1903.0615 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER