User:Boris Tsirelson/Sandbox1

=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–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 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.
 * 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").