(Redirected from Trivial topology)
A topological space is termed an indiscrete space if it satisfies the following equivalent conditions:
- It has an empty subbasis.
- It has a basis comprising only the whole space.
- The only open subsets are the whole space and the empty subset.
- The only closed subsets are the whole space and the empty subset.
- The space is either an empty space or its Kolmogorov quotient is a one-point space.
In some conventions, empty spaces are considered indiscrete. In other conventions, we exclude empty spaces from consideration.
Relation with other properties
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|ultraconnected space|||FULL LIST, MORE INFO|
|normal space|||FULL LIST, MORE INFO|
|regular space|||FULL LIST, MORE INFO|
|preregular space|||FULL LIST, MORE INFO|
|symmetric space|||FULL LIST, MORE INFO|