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  • Homomorphism (Algebra) 同态
  • Homeomorphism (Topology) 同胚
  • Isomorphism 同构

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  1. Math
  2. Topology

Concepts

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Last updated 2 years ago

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Homomorphism (Algebra) 同态

  • Wikipedia: homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).

  • Roughly speaking, homomorphism = mapping (one direction, not necessarily surjective) + preserving mathematical structures.

  • Of the same type:

    • unit quaternions are isomorphic to SU(2) -> not rigorous, because quaternion by itself is a number system, not a group

    • the group of unit quaternions (with multiplication as the binary operation, implicitly) is isomorphic to SU(2) -> good

Homeomorphism (Topology) 同胚

  • Not to be confused with homomorphism in algebra, which does not imply bijection.

  • Homeomorphism is a concept in topology, which implies bijection + preserving mathematical structures. It is an isomorphism of topological spaces.

Isomorphism 同构

  • Wikipedia: Isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping.

  • Roughly speaking, isomorphism = bijection + preserving mathematical structures.

In various areas of mathematics, isomorphisms have received specialized names, depending on the type of structure under consideration. For example:

  • An is an isomorphism of .

  • A is an isomorphism of .

  • A is an isomorphism of spaces equipped with a , typically .

  • A is an automorphism of a .

  • In , isomorphisms and automorphisms are often called , for example , , .

More concepts and topics to discuss:

  • Charts, Atlas, Connected, Simply connected, Projective space

  • The symbol ≅ can in principle be used to designate an isomorphism in any category (e.g., isometric, diffeomorphic, homeomorphic, linearly isomorphic, etc.)

isometry
metric spaces
homeomorphism
topological spaces
diffeomorphism
differential structure
differentiable manifolds
permutation
set
geometry
transformations
rigid transformations
affine transformations
projective transformations