Resolvable 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
History
Origin
The term resolvable space was introduced by E. Hewitt in 1943.
Definition
A topological space is said to be resolvable if it has two disjoint dense subsets. Note that since any subset containing a dense subset is dense, this is equivalent to saying that it is expressible as a union of two disjoint dense subsets.
Note that by this definition, the one-point space is not a resolvable space, but the empty space is a resolvable space.
Examples
The real numbers form a resolvable space. The rationals and irrationals both form disjoint dense subsets.
Relation with other properties
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| almost resolvable space | ||||
| dense-in-itself space | there are no isolated points | resolvable implies dense-in-itself | dense-in-itself not implies resolvable |
Metaproperties
| Metaproperty name | Satisfied? | Proof | Statement with symbols |
|---|---|---|---|
| open subspace-closed property of topological spaces | Yes | resolvability is open subspace-closed |
References
- A problem of set-theoretic topology by E. Hewitt, Duke Math J., 1943