Schröder-Bernstein property

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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 also Y is similar to a part of X then X and Y are similar (to each other).

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 Schröder–Bernstein (or Cantor–Schröder–Bernstein, or Cantor–Bernstein) 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.

A Schröder–Bernstein property is a joint property of

  • a class of objects,
  • a binary relation "be a part of",
  • a binary relation "be similar".

Instead of the relation "be a part of" one may use a binary relation "be embeddable into" interpreted as "be similar to some part of". Then a Schröder–Bernstein property takes the following form.

If X is embeddable into Y and Y is embeddable into X then X and Y are similar.

The same in the language of category theory:

If objects X, Y are such that X injects into Y (more formally, there exists a monomorphism from X to Y) and also Y injects into X then X and Y are isomorphic (more formally, there exists an isomorphism from X to Y).

A problem of deciding, whether a Schröder–Bernstein property (for a given class and two relations) holds or not, is called a Schröder–Bernstein problem. A theorem that states a Schröder–Bernstein property (for a given class and two relations), thus solving the Schröder–Bernstein problem in the affirmative, is called a Schröder–Bernstein theorem (for the given class and two relations), not to be confused with the classical (Cantor–)Schröder–Bernstein theorem mentioned above.

The Schroeder–Bernstein theorem for measurable spaces[1] states the Schröder–Bernstein property for

  • the class of measurable spaces,
  • "a part" is interpreted as a measurable subset treated as a measurable space,
  • "similar" is interpreted as isomorphic.

It has a noncommutative counterpart, the Schroeder–Bernstein theorem for operator algebras.

Two Schroeder–Bernstein theorems for Banach spaces are well-known. Both use

  • the class of Banach spaces, and
  • "similar" is interpreted as linearly homeomorphic.

They differ in the treatment of "part". One theorem[2] treats "part" as a subspace; the other theorem[3] treats "part" as a complemented subspace.

Many other Schröder–Bernstein problems are discussed by informal groups of mathematicians (see the external links page).

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.