User:TakuyaMurata/Automatic continuity

In mathematics, automatic continuity of a linear operator is the continuity that arises purely from the algebraic structures of the domain and the codomain of the operator.

The important example is a homomorphism between Banach algebras. Suppose there is a homeomorphism between $A$ . The map is continuity. In the view of the open mapping theorem, the inverse of the map is also continuous. We thus conclude that $A$ and $B$ are homeomorphic.