Sober space: Difference between revisions
m (1 revision) |
|||
| Line 11: | Line 11: | ||
===Stronger properties=== | ===Stronger properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Weaker than::Hausdorff space]] || any two distinct points can be separated by disjoint open subsets || || || {{intermediate notions|sober space|Hausdorff space}} | |||
|- | |||
| [[Weaker than::sober T1 space]] || the irreducible closed subsets are precisely the singleton subsets || || || {{intermediate notions short|sober space|sober T1 space}} | |||
|} | |||
Revision as of 00:48, 5 January 2017
This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces
Definition
Symbol-free definition
A topological space is said to be sober if the only irreducible closed subsets are the closures of one-point sets.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Hausdorff space | any two distinct points can be separated by disjoint open subsets | Template:Intermediate notions | ||
| sober T1 space | the irreducible closed subsets are precisely the singleton subsets | Sober T0 space|FULL LIST, MORE INFO |