User:Boris Tsirelson/Sandbox1

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.

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.

External links

Theme and variations: Schroeder-Bernstein

When does Cantor Bernstein hold?