User:Boris Tsirelson/Sandbox1: Difference between revisions
imported>Boris Tsirelson |
imported>Boris Tsirelson |
||
| Line 24: | Line 24: | ||
* a binary relation "be similar". | * a binary relation "be similar". | ||
Instead of the second relation 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. | Instead of the second relation 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 | :If ''X'' is embeddable into ''Y'' and ''Y'' is embeddable into ''X'' then ''X'' and ''Y'' are similar. | ||
A theorem that states a Schröder–Bernstein property (for a given class and two relations) is often called a Schröder–Bernstein theorem (for the given class and two relations), not to be confused with the classical (Cantor–)Bernstein–Schröder theorem mentioned above. | A theorem that states a Schröder–Bernstein property (for a given class and two relations) is often called a Schröder–Bernstein theorem (for the given class and two relations), not to be confused with the classical (Cantor–)Bernstein–Schröder theorem mentioned above. | ||
Revision as of 05:53, 2 September 2010
Schröder–Bernstein property
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 at the same time Y is similar to a part of X then X and Y are similar.
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 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
- X and Y 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 second relation 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.
A theorem that states a Schröder–Bernstein property (for a given class and two relations) is often called a Schröder–Bernstein theorem (for the given class and two relations), not to be confused with the classical (Cantor–)Bernstein–Schröder theorem mentioned above.
Notes
References
Srivastava, S.M. (1998), A Course on Borel Sets, Springer. See Proposition 3.3.6 (on page 96), and the first paragraph of Section 3.3 (on page 94).
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.