Schröder-Bernstein property: Difference between revisions
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In the classical Schröder–Bernstein (or Cantor–Schröder–Bernstein, or Cantor–Bernstein) theorem, | In the classical Schröder–Bernstein (or Cantor–Schröder–Bernstein, or Cantor–Bernstein) theorem, | ||
* | * objects are [[Set (mathematics)|sets]] (maybe infinite), | ||
* "a part" is interpreted as a [[subset]], | * "a part" is interpreted as a [[subset]], | ||
* "similar" is interpreted as [[Bijective function#Bijections and the concept of cardinality|equinumerous]]. | * "similar" is interpreted as [[Bijective function#Bijections and the concept of cardinality|equinumerous]]. | ||
Not all statements of this form are true. For example, assume that | Not all statements of this form are true. For example, assume that | ||
* | * objects are [[triangle]]s, | ||
* "a part" means a triangle inside the given triangle, | * "a part" means a triangle inside the given triangle, | ||
* "similar" is interpreted as usual in elementary geometry: triangles related by a dilation (in other words, "triangles with the same shape up to a scale factor", or equivalently "triangles with the same angles"). | * "similar" is interpreted as usual in elementary geometry: triangles related by a dilation (in other words, "triangles with the same shape up to a scale factor", or equivalently "triangles with the same angles"). | ||
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A problem of deciding, whether a Schröder–Bernstein property (for a given class and two relations) holds or not, is called a Schröder–Bernstein problem. A theorem that states a Schröder–Bernstein property (for a given class and two relations), thus solving the Schröder–Bernstein problem in the affirmative, is called a Schröder–Bernstein theorem (for the given class and two relations), not to be confused with the classical (Cantor–)Schröder–Bernstein theorem mentioned above. | A problem of deciding, whether a Schröder–Bernstein property (for a given class and two relations) holds or not, is called a Schröder–Bernstein problem. A theorem that states a Schröder–Bernstein property (for a given class and two relations), thus solving the Schröder–Bernstein problem in the affirmative, is called a Schröder–Bernstein theorem (for the given class and two relations), not to be confused with the classical (Cantor–)Schröder–Bernstein theorem mentioned above. | ||
The Schröder–Bernstein theorem for [[measurable space]]s<ref>{{harvnb|Srivastava|1998}}, see Proposition 3.3.6 (on page 96), and the first paragraph of Section 3.3 (on page 94).</ref> states the Schröder–Bernstein property for | The Schröder–Bernstein theorem for [[measurable space]]s<ref>{{harvnb|Srivastava|1998}}, see Proposition 3.3.6 (on page 96), and the first paragraph of Section 3.3 (on page 94).</ref> states the Schröder–Bernstein property for the following case: | ||
* | * objects are measurable spaces, | ||
* "a part" is interpreted as a measurable subset treated as a measurable space, | * "a part" is interpreted as a measurable subset treated as a measurable space, | ||
* "similar" is interpreted as isomorphic. | * "similar" is interpreted as isomorphic. | ||
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Two Schröder–Bernstein theorems for [[Banach space]]s are well-known. Both use | Two Schröder–Bernstein theorems for [[Banach space]]s are well-known. Both use | ||
* | * objects are Banach spaces, | ||
* "a part" is interpreted as a subspace<ref>{{harvnb|Casazza|1989}}</ref> or a complemented subspace<ref>{{harvnb|Gowers|1996}}</ref>, | |||
* "similar" is interpreted as linearly homeomorphic. | * "similar" is interpreted as linearly homeomorphic. | ||
Many other Schröder–Bernstein problems related to various [[space (mathematics)|spaces]] and algebraic structures (groups, rings, fields etc) are discussed by informal groups of mathematicians (see the [[Schröder–Bernstein property/External Links|external links page]]). | Many other Schröder–Bernstein problems related to various [[space (mathematics)|spaces]] and algebraic structures (groups, rings, fields etc) are discussed by informal groups of mathematicians (see the [[Schröder–Bernstein property/External Links|external links page]]). | ||
Revision as of 14:24, 2 September 2010
A mathematical property is said to be a Schröder–Bernstein (or Cantor–Schröder–Bernstein, or Cantor–Bernstein) property if it is formulated in the following form.
- If X is similar to a part of Y and also Y is similar to a part of X then X and Y are similar (to each other).
In order to be specific one should decide
- what kind of mathematical objects are X and Y,
- what is meant by "a part",
- what is meant by "similar".
In the classical Schröder–Bernstein (or Cantor–Schröder–Bernstein, or Cantor–Bernstein) theorem,
- objects are sets (maybe infinite),
- "a part" is interpreted as a subset,
- "similar" is interpreted as equinumerous.
Not all statements of this form are true. For example, assume that
- objects are triangles,
- "a part" means a triangle inside the given triangle,
- "similar" is interpreted as usual in elementary geometry: triangles related by a dilation (in other words, "triangles with the same shape up to a scale factor", or equivalently "triangles with the same angles").
Then the statement fails badly: every triangle X evidently is similar to some triangle inside Y, and the other way round; however, X and Y need no be similar.
A Schröder–Bernstein property is a joint property of
- a class of objects,
- a binary relation "be a part of",
- a binary relation "be similar".
Instead of the relation "be a part of" one may use a binary relation "be embeddable into" interpreted as "be similar to some part of". Then a Schröder–Bernstein property takes the following form.
- If X is embeddable into Y and Y is embeddable into X then X and Y are similar.
The same in the language of category theory:
- If objects X, Y are such that X injects into Y (more formally, there exists a monomorphism from X to Y) and also Y injects into X then X and Y are isomorphic (more formally, there exists an isomorphism from X to Y).
A problem of deciding, whether a Schröder–Bernstein property (for a given class and two relations) holds or not, is called a Schröder–Bernstein problem. A theorem that states a Schröder–Bernstein property (for a given class and two relations), thus solving the Schröder–Bernstein problem in the affirmative, is called a Schröder–Bernstein theorem (for the given class and two relations), not to be confused with the classical (Cantor–)Schröder–Bernstein theorem mentioned above.
The Schröder–Bernstein theorem for measurable spaces[1] states the Schröder–Bernstein property for the following case:
- objects are measurable spaces,
- "a part" is interpreted as a measurable subset treated as a measurable space,
- "similar" is interpreted as isomorphic.
It has a noncommutative counterpart, the Schröder–Bernstein theorem for operator algebras.
Two Schröder–Bernstein theorems for Banach spaces are well-known. Both use
- objects are Banach spaces,
- "a part" is interpreted as a subspace[2] or a complemented subspace[3],
- "similar" is interpreted as linearly homeomorphic.
Many other Schröder–Bernstein problems related to various spaces and algebraic structures (groups, rings, fields etc) are discussed by informal groups of mathematicians (see the external links page).
Notes
- ↑ Srivastava 1998, see Proposition 3.3.6 (on page 96), and the first paragraph of Section 3.3 (on page 94).
- ↑ Casazza 1989
- ↑ Gowers 1996
References
Srivastava, S.M. (1998), A Course on Borel Sets, Springer.
Gowers, W.T. (1996), "A solution to the Schroeder-Bernstein problem for Banach spaces", Bull. London Math. Soc. 28: 297–304.
Casazza, P.G. (1989), "The Schroeder-Bernstein property for Banach spaces", Contemp. Math. 85: 61–78.