Separable space
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
This is a variation of countability. View other variations of countability
Definition
Symbol-free definition
A topological space is said to be separable if it has a countable dense subset.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| second-countable space | ||||
| Polish space | ||||
| hereditarily separable space | ||||
| sub-Euclidean space | ||||
| manifold |
Metaproperties
Products
This property of topological spaces is closed under taking finite products
A finite product of separable spaces is separable. The countable dense subset that we take for the product is the Cartesian product of countable dense subsets for each.
Hereditariness on open subsets
This property of topological spaces is hereditary on open subsets, or is open subspace-closed. In other words, any open subset of a topological space having this property, also has this property
Any open subset of a separable space is separable. Indeed, given a countable dense subset for the whole space, we intersect it with the open subset to get a countable dense subset of the open subset.
Hereditariness
This property of topological spaces is not hereditary on all subsets
It is not true in general that a subspace of a separable space is separable. A counterexample is the antidiagonal in the Sorgenfrey plane.
Effect of property operators
The hereditarily operator
Applying the hereditarily operator to this property gives: hereditarily separable space
References
Textbook references
- Topology (2nd edition) by James R. MunkresMore info, Page 192 (definition in paragraph)