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=Schröder–Bernstein property=
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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,
* ''X'' and ''Y'' are [[Set (mathematics)|sets]] (maybe infinite),
* "a part" is interpreted as a [[subset]],
* "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
* ''X'' and ''Y'' are [[triangle]]s,
* "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 Schroeder–Bernstein theorem for [[measurable space]]s<ref>{{harvnb|Srivastava|1998}}</ref> states the Schröder–Bernstein property for
* the class of 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 Schroeder–Bernstein theorem for operator algebras.
 
Two Schroeder–Bernstein theorems for [[Banach space]]s are well-known. Both use
* the class of Banach spaces, and
* "similar" is interpreted as linearly homeomorphic.
They differ in the treatment of "part". One theorem<ref>{{harvnb|Casazza|1989}}</ref> treats "part" as a subspace; the other theorem<ref>{{harvnb|Gowers|1996}}</ref> treats "part" as a complemented subspace.
 
Many other Schröder–Bernstein problems are discussed by informal groups of mathematicians (see the external links page).
 
==Notes==
{{reflist}}
 
==References==
 
{{Citation
| last = Srivastava
| first = S.M.
| title = A Course on Borel Sets
| year = 1998
| publisher = Springer
}}. See Proposition 3.3.6 (on page 96), and the first paragraph of Section 3.3 (on page 94).
 
{{Citation
| last = Gowers
| first = W.T.
| year = 1996
| title = A solution to the Schroeder-Bernstein problem for Banach spaces
| journal = Bull. London Math. Soc.
| volume = 28
| pages = 297–304
| url = http://blms.oxfordjournals.org/content/28/3/297
}}.
 
{{Citation
| last = Casazza
| first = P.G.
| year = 1989
| title = The Schroeder-Bernstein property for Banach spaces
| journal = Contemp. Math.
| volume = 85
| pages = 61–78
| url = http://www.ams.org/mathscinet-getitem?mr=983381
}}.
 
==External links==
 
[http://sbseminar.wordpress.com/2007/10/30/theme-and-variations-schroeder-bernstein/ Theme and variations: Schroeder-Bernstein]
 
[http://mathoverflow.net/questions/1058/when-does-cantor-bernstein-hold When does Cantor Bernstein hold?]

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