# Concepts

**Homomorphism (Algebra) **同态

**Homomorphism (Algebra)**同态

Wikipedia: homomorphism is a structure-preserving map between two algebraic structures

(such as two groups, two rings, or two vector spaces).**of the same type**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 isometry is an isomorphism of metric spaces.

**A****homeomorphism****is an isomorphism of****topological spaces****.**A diffeomorphism is an isomorphism of spaces equipped with a differential structure, typically differentiable manifolds.

A permutation is an automorphism of a set.

In geometry, isomorphisms and automorphisms are often called transformations, for example rigid transformations, affine transformations, projective transformations.

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.)

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