User:Boris Tsirelson/Sandbox1: Difference between revisions
imported>Boris Tsirelson |
imported>Boris Tsirelson |
||
Line 13: | Line 13: | ||
* "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, | Not all statements of this form are true. For example, assume that | ||
* ''X'' and ''Y'' are [[triangle]]s, | * ''X'' and ''Y'' are [[triangle]]s, | ||
* "a part" means a triangle inside the given triangle, | * "a part" means a triangle inside the given triangle, |
Revision as of 05:39, 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 Cantor–Bernstein–Schröder 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.
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.