Homeomorphism

Example: Visualization of a homeomorphism between Cantor spaces . Homeomorphism from in to . The colors indicate how subspaces of sequences with a common prefix are mapped onto one another.

{\ displaystyle 3 ^ {\ omega}}

{\ displaystyle 2 ^ {\ omega}}

A homeomorphism (sometimes also homeomorphism based on the English term homeomorphism , but not to be confused with homomorphism ) is a central term in the mathematical sub-area of topology . It designates a bijective , continuous mapping between two topological spaces , the reverse mapping of which is also continuous. The continuity property depends on the topological spaces considered.

Two topological spaces are called homeomorphic (also topologically equivalent) if they can be converted into one another by a homeomorphism (also topological mapping or topological isomorphism); they are in the same homeomorphism class and, from a topological point of view, are of the same kind. The topology examines properties that are invariant under homeomorphisms .

One can clearly imagine a homeomorphism as stretching, compressing, bending, distorting, twisting an object; Cutting is only allowed if the parts are later put back together exactly at the cut surface.

definition

${\ displaystyle X}$ and be topological spaces . A mapping is a homeomorphism if and only if: ${\ displaystyle Y}$ ${\ displaystyle f \ colon X \ rightarrow Y}$

${\ displaystyle f}$ is bijective
${\ displaystyle f}$ is steady
the inverse function is also continuous. ${\ displaystyle f ^ {- 1}}$

Homeomorphisms can be characterized as follows: If and are topological spaces, then for a bijective, continuous mapping they are equivalent: ${\ displaystyle X}$ ${\ displaystyle Y}$ ${\ displaystyle f \ colon X \ to Y}$

${\ displaystyle f}$ is a homeomorphism.
${\ displaystyle f}$ is an open figure .
${\ displaystyle f}$ is a completed figure.

Examples

Every open disk (with a positive radius) is homeomorphic to every open square (with a positive side) in the Euclidean plane . A circular disk can therefore be transformed into a square by bending and distorting it without cutting, and vice versa. ${\ displaystyle \ mathbb {R} ^ {2}}$

The open interval is homeomorphic to the space of all real numbers. Any open interval can be easily distorted to infinity. A homeomorphism that mediates this for is for example ${\ displaystyle \ left] 0.1 \ right [}$ ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle \ left] 0.1 \ right [}$

{\ displaystyle {\ begin {aligned} f \ colon \ left] 0.1 \ right [& \ to \ mathbb {R} \\ x & \ mapsto \ tan \ left (\ left (x - {\ tfrac {1} {2}} \ right) \ cdot \ pi \ right) \ end {aligned}}}

The product space of the unit circle with itself is homeomorphic to the two-dimensional torus , i.e. to the shape of a bicycle inner tube. For a homeomorphism that conveys this, a point on the first circle is first assigned a point on the rim of the bicycle tire, then a point on the second circle is assigned a point on the tire cross-section adjacent to the rim point. ${\ displaystyle \ mathbb {S} ^ {1} \ times \ mathbb {S} ^ {1}}$

Every isomorphism in the sense of functional analysis is a homeomorphism.

Importance of reversibility

The third condition of the continuity of the inverse function is essential. For example, consider the function ${\ displaystyle f ^ {- 1}}$

{\ displaystyle {\ begin {aligned} f \ colon \ left [0,2 \ pi \ right [& \ to \ mathbb {S} ^ {1} \\ x & \ mapsto \ left (\ cos (x), \ sin (x) \ right) \ end {aligned}}}

This function is continuous and bijective, but not a homeomorphism. The inverse function maps points close to to widely spaced numbers near and ; vividly, the circle would be torn at the point and then rolled flat to form an interval. ${\ displaystyle f ^ {- 1}}$ ${\ displaystyle (1,0)}$ ${\ displaystyle 0}$ ${\ displaystyle 2 \ pi}$ ${\ displaystyle (1,0)}$

If one restricts oneself to certain types of topological spaces, then the continuity of the inverse mapping of a bijection already follows from the continuity of . For example, a continuous bijection between compact Hausdorff spaces is already a homeomorphism. The following serves to prove this statement ${\ displaystyle f}$ ${\ displaystyle f}$

sentence: If there is a compact and a Hausdorff topological space, then every continuous bijective mapping is a homeomorphism. ${\ displaystyle X}$ ${\ displaystyle Y}$ ${\ displaystyle f \ colon X \ to Y}$

proof: Be the reverse figure and complete, it is to show that is complete. As a self-contained subset of a compact is compact. Since continuous images of compact quantities are compact again, is compact. Since compact sets are closed in Hausdorff spaces , what ends the proof is closed. ${\ displaystyle g \ colon Y \ to X}$ ${\ displaystyle A \ subseteq X}$ ${\ displaystyle g ^ {- 1} (A)}$ ${\ displaystyle A}$ ${\ displaystyle g ^ {- 1} (A) = f (A)}$ ${\ displaystyle g ^ {- 1} (A)}$

properties

If two topological spaces are homeomorphic, then they have exactly the same topological properties, these are properties that can only be expressed by the underlying set and the open or closed sets defined on it. This is because a homeomorphism is by definition a bijection between the underlying sets and between the systems of open sets. Examples of such properties are compactness , cohesion , separation properties and many more. Evidence that it is a topological property can be difficult, especially if the original definition uses additional structures. An example of such a property is metrizability , here the Bing-Nagata-Smirnow theorem shows that it is a topological property. Eberlein compactness is another non-trivial example.

But there are also properties of certain spaces that are not retained in homeomorphisms, for example the completeness of metric spaces. The plane and the open circular disk with the standard metric are homeomorphic with respect to the topologies defined by the metric, the former is complete, the latter is not. Completeness is therefore not a topological property, it is not retained in homeomorphisms.

Local homeomorphism

A continuous mapping between topological spaces is called local homeomorphism if there is an open neighborhood of for every point such that ${\ displaystyle f}$ ${\ displaystyle X, Y}$ ${\ displaystyle a \ in X}$ ${\ displaystyle U \ subseteq X}$ ${\ displaystyle a}$

${\ displaystyle f (U) \ subseteq Y}$ an open environment of educates and ${\ displaystyle f (a)}$
${\ displaystyle f | _ {U} \ colon U \ rightarrow f (U)}$ is a homeomorphism.

Every homeomorphism is also a local homeomorphism, but the converse does not apply, as the following example shows: The mapping is not bijective, but a local homeomorphism, since the derivative of nowhere does not vanish. ${\ displaystyle f \ colon \ mathbb {C} \ setminus \ left \ {0 \ right \} \ rightarrow \ mathbb {C}, \, x \ mapsto x ^ {2}}$ ${\ displaystyle f}$