Resolvable space: Difference between revisions
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===Weaker properties=== | ===Weaker properties=== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Stronger than::almost resolvable space]] || || || || | |||
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| [[Stronger than::space with no isolated points]] || there are no [[isolated point]]s || [[resolvable implies no isolated points]] || [[no isolated points not implies resolvable]] || | |||
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==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'' | ||
Revision as of 20:13, 26 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 | ||||
| space with no isolated points | there are no isolated points | resolvable implies no isolated points | no isolated points not implies resolvable |
References
- A problem of set-theoretic topology by E. Hewitt, Duke Math J., 1943