In general topology, a homeomorphism is a map between spaces that preserves all topological properties. Intuitively, given some sort of geometric object, a topological property is a property of the object that remains unchanged after the object has been stretched or deformed in some way.
For example, a space is called path-connected if any two points in can be joined by a continuous curve. In symbols, path-connectedness requires that for any , , there is a continuous function such that and . Path-connectedness should be considered a topological property, since stretching would simply stretch as well, and the two points in question would remain connected by a path in the new, stretched space.
Formally, a homeomorphism between two topological spaces and is a bijection such that is continuous and is also continuous. Then, topological properties are defined to be those properties of topological spaces that are preserved under homeomorphism. Recall that continuous maps are essentially those which send points close to one another in the domain to points close to one another in the codomain.
For any , the open interval is homeomorphic to .
To show this, first note that is homeomorphic to via the map . This is a homeomorphism since it is a continuous bijection whose inverse, , is also continuous.
Thus, it will suffice to exhibit a homeomorphism . The map given by will do the trick, since it is a continuous bijection whose inverse, , is also continuous.
Stereographic projection is an important homeomorphism between the plane and the -sphere minus a point. The -sphere is the set of points such that . Let denote the -sphere minus its north pole, i.e. the point .
There exists a homeomorphism , which can be described as follows. First, identify the set with ; the map given by is a homeomorphism.
For a point , let denote the unique point in such that the intersection of the segment and is . In coordinates, this map is precisely
Via similarly defined maps, one can show that the -sphere minus a point i.e., is homeomorphic to .
The Cartesian plane is not homeomorphic to the real line . This can be proven by contradiction.
Suppose there is a homeomorphism . Choose , and consider . If is a homeomorphism, then restricts to a homeomorphism between and . But is path-connected, while is not! Thus, cannot be a homeomorphism.