Indiscrete space: Difference between revisions
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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 [[coarser topology|coarsest possible topology]] on the set. | 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 [[coarser topology|coarsest possible topology]] on the set. | ||
In some conventions, empty spaces | In some conventions, empty spaces are considered indiscrete. In other conventions, we exclude empty spaces from consideration. | ||
==Relation with other properties== | ==Relation with other properties== | ||
Latest revision as of 18:23, 28 January 2012
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 |