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 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 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 treats "part" as a subspace; the other theorem treats "part" as a complemented subspace.

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

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.