Homeomorphism

This word must not be confused with homomorphism.

In topology, two geometrical objects (or "spaces") are called homeomorphic if, roughly speaking, the first can be deformed into the second by stretching and bending; cutting is also allowed, but only if the two parts are later glued back together along exactly the same cut. For example, a square and a circle are homeomorphic. A hollow sphere containing a smaller solid ball is homeomorphic to a hollow cube with a solid cube outside of it. If two objects are homeomorphic, one can find a continuous function which maps points from the first object to corresponding points of the second object, and vice versa. Such a function is called a homeomorphism; intuitively, it maps points in the first object that are "close together" to points in the second object that are close together, and points in the first object that are not close together to points in the second object that are not close together. Topology is the study of those properties of objects that do not change when homeomorphisms are applied.

For a formal definition, suppose X and Y are topological spaces, and f is a function from X to Y. Then f is a homeomorphism iff all the following hold:

f is a bijection,
f is continuous,
the inverse function f^-1 is continuous.

If there exists a homeomorphism f : X -> Y, then Y is said to be homeomorphic to X (or to be a homeomorph of X). In this case, Y is also homeomorphic to X, since f^-1 is a homeomorphism, and we say that X and Y belong to the same homeomorphism class.

For example, the unit circle S¹ and the unit square in R² are homeomorphic. The open interval (-1, 1) is homeomorphic to the real numbers R. The product space S¹ × S¹ and the two-dimensional torus are homeomorphic.

The third requirement, that f^-1 be continuous, is essential. Consider for instance the function f : [0, 2π) -> S¹ defined by f(φ) = (cos(φ), sin(φ)). This function is bijective and continuous, but not a homeomorphism.

If two spaces are homeomorphic then they have exactly the same topological properties. For example, if one of them is compact, then the other is as well; if one of them is connected, then the other is as well; if one of them is Hausdorff, then the other is as well; their homology groups will coincide. Note however that this does not extend to properties defined via a metric; there are metric spaces which are homeomorphic even though one of them is complete and the other is not.

Homeomorphisms are the isomorphisms in the category of all topological spaces. As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all homeomorphisms X → X forms a group.

Informal discussion

The intuitive criterion of stretching, bending, cutting and gluing back together takes a certain amount of practice to apply correctly--it may not be obvious from the description above that deforming a line segment to a point is impermissible, for instance. It is thus important to realize that it is the formal definition given above that counts.

This characterization of a homeomorphism often leads to confusion with the concept of homotopy, which is actually defined as a continuous deformation, but from one function to another, rather than one space to another. In the case of a homeomorphism, envisioning a continuous deformation is a mental tool for keeping track of which points on space X correspond to which points on Y -- one just follows them as X deforms. In the case of homotopy, the continuous deformation from one map to the other is of the essence, and it is also less restrictive, since none of the maps involved need to be one-to-one or onto. Homotopy does lead to a relation on spaces: homotopy equivalence.

There is a name for the kind of deformation involved in visualizing a homeomorphism. It is (except when cutting and regluing are required) an isotopy between the identity map on X and the homeomorphism from X to Y.