Almost discrete space
Definition
A topological space is termed almost discrete if it satisfies the following equivalent conditions:
- Every open subset is a closed subset
- Every closed subset is an open subset
- It is both an Alexandrov space (i.e., it is finitely generated) and a zero-dimensional space (i.e., it has a basis of clopen subsets).
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
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| discrete space | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Alexandrov space | ||||
| zero-dimensional space |