User:Boris Tsirelson/Sandbox1

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

Bourbaki, Nicolas (1968), Elements of mathematics: Theory of sets, Hermann (original), Addison-Wesley (translation).

Casacuberta, C & M Castellet, eds. (1992), Mathematical research today and tomorrow: Viewpoints of seven Fields medalists, Lecture Notes in Mathematics, vol. 1525, Springer-Verlag, ISBN 3-540-56011-4.

Feynman, Richard (1995), The character of physical law (twenty second printing ed.), the MIT press, ISBN 0 262 56003 8.

Gowers, Timothy, ed. (2008), The Princeton companion to mathematics, Princeton University Press, ISBN 978-0-691-11880-2.

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), "[http: The Schroeder-Bernstein property for Banach spaces]", Contemp. Math. 85: 61–78. (Also here.)