# Indiscrete space

From Topospaces

(Redirected from Trivial topology)

## Definition

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.

For any set, there is a unique topology on it making it an indiscrete space. This topology is called the **indiscrete topology** or the **trivial topology**. It is the coarsest possible topology on the set.

In some conventions, empty spaces are considered indiscrete. In other conventions, we exclude empty spaces from consideration.

## Relation with other properties

### Weaker 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 | |||

contractible space |