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Homeomorphism

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Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that 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. The word homeomorphism comes from the Greek words ὅμοιος (homoios) = similar and μορφή (morphē) = shape, form.

Contents

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 torus are not. An often-repeated mathematical joke is that topologists can't tell the difference between a coffee cup and a donut, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in a cup's handle.

Definition

A function f: X → Y between two topological spaces (X, TX) and (Y, TY) 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 open interval (a, b) is homeomorphic to the real numbers R for any a < b. (In this case, a bicontinuous forward mapping is given by f(x) = 1/(a − x) + 1/(b − x) while other such mappings are given by scaled and translated versions of the tan or arg tanh functions).
  • The unit 2-disc D2 and the unit square in R2 are homeomorphic; since the unit disc can be deformed into the unit square. An example of a bicontinuous mapping from the square to the disc is, in polar coordinates, ( ρ , θ ) ↦ ( ρ max ( | cos ⁡ θ | , | sin ⁡ θ | ) , θ ) .
  • The graph of a differentiable function is homeomorphic to the domain of the function.
  • A differentiable parametrization of a curve is an homeomorphism between the domain of the parametrization and the curve.
  • A chart of a manifold is an homeomorphism between an open subset of the manifold and an open subset of a Euclidean space.
  • The stereographic projection is a homeomorphism between the unit sphere in R3 with a single point removed and the set of all points in R2 (a 2-dimensional plane).
  • If G is a topological group, its inversion map x ↦ x − 1 is a homeomorphism. Also, for any x ∈ G , the left translation y ↦ x y , the right translation y ↦ y x , and the inner automorphism y ↦ x y x − 1 are homeomorphisms.

    • Non-examples

  • Rm and Rn are not homeomorphic for m ≠ n.
  • The Euclidean real line is not homeomorphic to the unit circle as a subspace of R2, since the unit circle is compact as a subspace of Euclidean R2 but the real line is not compact.

    • 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 homotopy & homology groups will coincide. Note however that this does not extend to properties defined via a metric; there are metric spaces that 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 a 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.

    References

    Homeomorphism Wikipedia
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