User:Boris Tsirelson/Sandbox1: Difference between revisions
imported>Boris Tsirelson No edit summary |
imported>Boris Tsirelson No edit summary |
||
Line 1: | Line 1: | ||
= | =Schröder–Bernstein property= | ||
The general idea of the Cantor–Bernstein–Schroeder theorem and related results may be formulated as follows. 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 | The general idea of the Cantor–Bernstein–Schroeder theorem and related results may be formulated as follows. 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 |
Revision as of 04:03, 2 September 2010
Schröder–Bernstein property
The general idea of the Cantor–Bernstein–Schroeder theorem and related results may be formulated as follows. 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–Schroeder 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, let
- 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.