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 $\backslash \{\; p\; \backslash in\; \backslash textrm\{Spec\; \}\; A\; :\; p\; \backslash cap\; S\; =\; \backslash emptyset\; \backslash \}$.

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