## Proof That the Diagonal Cartesian Product of a Set With Itself is Homeomorphic to the Set

Theorem

The diagonal cartesian productof a set with itself is homeomorphic to the set.

Proof

The setis the diagonal of

We can define projectionsandso thatand

Sincefor every element offor every element ofagrees withonandare one to one and ontoso and D are homeomorphic.