Resolvable space: Difference between revisions
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==Definition== | ==Definition== | ||
A [[topological space]] is said to be '''resolvable''' if it has two disjoint [[dense subset]]s. 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== | ==Examples== | ||
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===Weaker properties=== | ===Weaker properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Stronger than::almost resolvable space]] || || || || | |||
|- | |||
| [[Stronger than::dense-in-itself space]] || there are no [[isolated point]]s || [[resolvable implies dense-in-itself]] || [[dense-in-itself not implies resolvable]] || | |||
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==Metaproperties== | |||
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! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols | |||
|- | |||
| [[satisfies metaproperty::open subspace-closed property of topological spaces]] || Yes || [[resolvability is open subspace-closed]] || | |||
|} | |||
==References== | ==References== | ||
* ''A problem of set-theoretic topology'' by E. Hewitt, ''Duke Math J., 1943'' | * ''A problem of set-theoretic topology'' by E. Hewitt, ''Duke Math J., 1943'' | ||
Latest revision as of 02:00, 27 January 2012
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