In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.
Formally,
A function
between two topological spaces
and
is called a homeomorphism if it has the following properties:
- is a bijection (one-to-one and onto),
- is continuous,
- the inverse function is continuous ( is an open mapping).
A function with these three properties is sometimes called
bicontinuous. If such a function exists, we say
and
are
homeomorphic.
A self-homeomorphism is a homeomorphism of a topological space and itself. The homeomorphisms form an
equivalence relation on the
class of all topological spaces. The resulting
equivalence classes are called
homeomorphism classes.
A continuous deformation between a coffee mug and a donut (torus) illustrating that they are
homeomorphic ( a continuous mapping with a continuous inverse function ).