Links:Schroeder-Bernstein property

Theme and variations: Schroeder-Bernstein Various Schröder–Bernstein problems are discussed in a group blog by 8 recent Berkeley mathematics Ph.D.

When does Cantor Bernstein hold? "Mathoverflow" discusses the question in terms of category theory: "Can we characterize Cantor-Bernsteiness in terms of other categorical properties?"

Wikipedia
Cantor–Bernstein–Schroeder theorem

Schroeder–Bernstein theorem for measurable spaces

Schröder–Bernstein theorems for operator algebras

Commutative von Neumann algebras