id: 06426593
dt: j
an: 2015c.00524
au: Chih, Tien
ti: Counterexamples to Cantor-Schroeder-Bernstein.
so: Math. Enthus. 11, No. 3, 475-484 (2014).
py: 2014
pu: Information Age Publishing (IAP), Charlotte, NC; University of Montana,
Department of Mathematical Sciences, Missoula, MT
la: EN
cc: E65 I95 K35 H45
ut: Cantor-Schroeder-Bernstein theorem; equivalence theorem; cardinality;
proofs; counterexamples; object isomorphisms; set theory; injection;
bijection; applications of mathematics to mathematics; vector spaces;
dimension theorem; division rings; linear monomorphisms; topological
spaces; homeomorphisms; unital rings; non-unital rings; groups; graphs
ci:
li: http://www.math.umt.edu/tmme/vol11no3/Tien_2.pdf
ab: Summary: The Cantor-Schroeder-Bernstein theorem states that any two sets
that have injections into each other have the same cardinality, i.e.
there is a bijection between them. Another way to phrase this is if two
sets $A$, $B$ have monomorphisms from $A$ to $B$ and $B$ to $A$, then
they are isomorphic in the setting of sets. One naturally wonders if
this may be extended to other commonly studied systems of sets with
structure and functions which preserve that structure. Given two
objects with injective structure preserving maps between them are the
structures of these objects the same? In other words, would these two
objects be isomorphic in their respective setting? We see that in
vector spaces, which are determined completely by their bases sets,
this is true. However, when the objects are graphs, groups, rings or
topological spaces, one may find counterexamples to such an extension.
This is interesting, as it contradicts the naive intuition that two
objects which are â€œsubobjects" of each other must be the same. In
this paper we provide some of these counterexamples.
rv: