Homeomorphism

Topological equivalence redirects here; see also topological equivalence (dynamical systems).

In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function (from the Greek words ὅμοιος (homoios) = similar and μορφή (morphē) = shape, form) is a continuous function between two topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces — that is, they are the mappings which preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same.

Roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a donut are not. An often-repeated joke is that topologists can't tell the coffee cup from which they are drinking from the donut they are eating, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.

Topology is the study of those properties of objects that do not change when homeomorphisms are applied. As Henri Poincaré famously said, mathematics is not the study of objects, but instead, the relations (isomorphisms for instance) between them.

Definition

A function f: X → Y between two topological spaces (X, T_X) and (Y, T_Y) is called a homeomorphism if it has the following properties:

f is a bijection (one-to-one and onto),
f is continuous,
the inverse function f⁻¹ is continuous (f is an open mapping).

A function with these three properties is sometimes called bicontinuous. If such a function exists, we say X and Y 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.

Examples

The unit 2-disc D² 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.

Every uniform isomorphism and isometric isomorphism is a homeomorphism.

Any 2-sphere with a single point removed is homeomorphic to the set of all points in R² (a 2-dimensional plane).

Let $A$ be a commutative ring with unity and let $S$ be a multiplicative subset of $A$ . Then Spec $(A_{S})$ is homeomorphic to $\{p\in {\textrm {Spec}}A:p\cap S=\emptyset \}$ .

$\mathbb {R} ^{n}$ and $\mathbb {R} ^{m}$ are not homeomorphic for $n\neq m$ .

An example of a continuous bijection that is not a homeomorphism is the map that takes the half-open interval $[0,1)$ and wraps it around the circle. In this case the inverse – although it exists – fails to be continuous. The preimages of certain sets which are actually open in the relative topology of the half-open interval are not open in the more natural topology of the circle (they are half-open intervals).

Notes

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.

Homeomorphisms are the isomorphisms in the category of topological spaces. As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms X → X forms a group, called the homeomorphism group of X, often denoted Homeo(X); this group can be given a topology, such as the compact-open topology, making it a topological group.

For some purposes, the homeomorphism group happens to be too big, but by means of the isotopy relation, one can reduce this group to the mapping class group.

Similarly, as usual in category theory, given two spaces that are homeomorphic, the space of homeomorphisms between, Homeo(X, Y) them is a torsor for the homeomorphism groups Homeo(X) and Homeo(Y), and given a specific homeomorphism between X and Y, all three sets are identified.

Properties

Two homeomorphic spaces share 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.

A homeomorphism is simultaneously an open mapping and a closed mapping, that is it maps open sets to open sets and closed sets to closed sets.

Every self-homeomorphism in $S^{1}$ can be extended to a self-homeomorphism of the whole disk $D^{2}$ (Alexander's Trick).

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.

External links

"Homeomorphism". PlanetMath.