Proof That the Diagonal Cartesian Product of a Set With Itself is Homeomorphic to the Set
Theorem
The diagonal cartesian product
of a set with itself is homeomorphic to the set.
Proof
The set
is the diagonal of
We can define projections
and
so that
and
Since
for every element of
for every element of
agrees with
onandare one to one and onto
so and D are homeomorphic.