Cantor–Bernstein theorem - Misplaced Pages

There are equally many countable order types and real numbers For the theorem that injections from A to B and from B to A imply a bijection between A and B, see Schröder–Bernstein theorem.

In set theory and order theory, the Cantor–Bernstein theorem states that the cardinality of the second type class, the class of countable order types, equals the cardinality of the continuum. It was used by Felix Hausdorff and named by him after Georg Cantor and Felix Bernstein. Cantor constructed a family of countable order types with the cardinality of the continuum, and in his 1901 inaugural dissertation Bernstein proved that such a family can have no higher cardinality.

References

Plotkin, J. M., ed. (2005). Hausdorff on Ordered Sets. History of Mathematics. Vol. 25. American Mathematical Society. p. 3. ISBN 9780821890516..

v t e Order theory
Topics Glossary Category
Key concepts	Binary relation Boolean algebra Cyclic order Lattice Partial order Preorder Total order Weak ordering
Results	Boolean prime ideal theorem Cantor–Bernstein theorem Cantor's isomorphism theorem Dilworth's theorem Dushnik–Miller theorem Hausdorff maximal principle Knaster–Tarski theorem Kruskal's tree theorem Laver's theorem Mirsky's theorem Szpilrajn extension theorem Zorn's lemma
Properties & Types (list)	Antisymmetric Asymmetric Boolean algebra topics Completeness Connected Covering Dense Directed (Partial) Equivalence Foundational Heyting algebra Homogeneous Idempotent Lattice Bounded Complemented Complete Distributive Join and meet Reflexive Partial order Chain-complete Graded Eulerian Strict Prefix order Preorder Total Semilattice Semiorder Symmetric Total Tolerance Transitive Well-founded Well-quasi-ordering (Better) (Pre) Well-order
Constructions	Composition Converse/Transpose Lexicographic order Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure
Topology & Orders	Alexandrov topology & Specialization preorder Ordered topological vector space Normal cone Order topology Order topology Topological vector lattice Banach Fréchet Locally convex Normed
Related	Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding Isomorphism Order type Ordered field Positive cone of an ordered field Ordered vector space Partially ordered Positive cone of an ordered vector space Riesz space Partially ordered group Positive cone of a partially ordered group Upper set Young's lattice

Category:

Order theory