> For the complete documentation index, see [llms.txt](https://wiki.hanzheteng.com/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://wiki.hanzheteng.com/math/topology/concepts.md). # Concepts ### **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 [isometry](https://en.wikipedia.org/wiki/Isometry) is an isomorphism of [metric spaces](https://en.wikipedia.org/wiki/Metric_space). * **A** [**homeomorphism**](https://en.wikipedia.org/wiki/Homeomorphism) **is an isomorphism of** [**topological spaces**](https://en.wikipedia.org/wiki/Topological_space)**.** * A [diffeomorphism](https://en.wikipedia.org/wiki/Diffeomorphism) is an isomorphism of spaces equipped with a [differential structure](https://en.wikipedia.org/wiki/Differential_structure), typically [differentiable manifolds](https://en.wikipedia.org/wiki/Differentiable_manifold). * A [permutation](https://en.wikipedia.org/wiki/Permutation) is an automorphism of a [set](https://en.wikipedia.org/wiki/Set_\(mathematics\)). * In [geometry](https://en.wikipedia.org/wiki/Geometry), isomorphisms and automorphisms are often called [transformations](https://en.wikipedia.org/wiki/Transformation_\(function\)), for example [rigid transformations](https://en.wikipedia.org/wiki/Rigid_transformation), [affine transformations](https://en.wikipedia.org/wiki/Affine_transformation), [projective transformations](https://en.wikipedia.org/wiki/Projective_transformation). 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.) --- # Agent Instructions This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com. ## Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter, and the optional `goal` query parameter: ``` GET https://wiki.hanzheteng.com/math/topology/concepts.md?ask=&goal= ``` `ask` is the immediate question: it should be specific, self-contained, and written in natural language. `goal` is optional and describes the broader end goal you are ultimately trying to accomplish on behalf of the user. GitBook uses it to tailor the answer towards what is most useful for that goal. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.