A homeomorphism is a continuous function between two topological spaces that has a continuous inverse function. They preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and are the same topologically.
A topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object. Thus, a square and a circle are homeomorphic to each other, but a sphere and torus are not. Any flat surface with no holes is homeomorphic to a square. Any three dimensional surface with a single hole is homeomorphic to the torus.
Topology is the study of those properties of objects that do not change when homeomorphisms are applied. Topology may be considered as the spacial properties of objects or the relationships between the spatial properties of different objects.
A function
between two topological spaces is called a homeomorphism if the following is true:
-
is a bijection (one-to-one and onto) - is continuous
-
the inverse function
is continuous (is an open mapping, that is, sends open sets to open sets).
If such a function exists, X and Y are homeomorphic. The homeomorphisms form an equivalence relation on the set of all topological spaces. The resulting equivalence classes are called homeomorphism classes.
Examples
-
The unit disk in
and the unit square inare homeomorphic. -
The open interval
is homeomorphic to the real numbers
-
The product space
and the two-dimensional torus are homeomorphic. -
The surface of a sphere in
with a single point removed is homeomorphic to the set of all points in(a 2-dimensional plane). The Riemann sphere with the operation of stereographic projection is an example. -
and
are not homeomorphic for
The number of dimensions of a space is a topological property, preserved by homeomorphisms.