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In mathematics and more specifically in topology, a homeomorphism (from Greek roots meaning "similar shape", named by Henri Poincaré), [2] [3] also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function.
In graph theory, two graphs and ′ are homeomorphic if there is a graph isomorphism from some subdivision of to some subdivision of ′.If the edges of a graph are thought of as lines drawn from one vertex to another (as they are usually depicted in diagrams), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if their diagrams are homeomorphic in the ...
A homeomorphism is a special case of a homotopy equivalence, in which g ∘ f is equal to the identity map id X (not only homotopic to it), and f ∘ g is equal to id Y. [7]: 0:53:00 Therefore, if X and Y are homeomorphic then they are homotopy-equivalent, but the opposite is not true. Some examples:
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).The word homomorphism comes from the Ancient Greek language: ὁμός (homos) meaning "same" and μορφή (morphe) meaning "form" or "shape".
A homomorphism from the flower snark J 5 into the cycle graph C 5. It is also a retraction onto the subgraph on the central five vertices. Thus J 5 is in fact homomorphically equivalent to the core C 5. In the mathematical field of graph theory, a graph homomorphism is a mapping between two graphs that respects their structure
The standard torus is homogeneous under its diffeomorphism and homeomorphism groups, and the flat torus is homogeneous under its diffeomorphism, homeomorphism, and isometry groups. In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group.
This homomorphism is "natural in ", i.e., it defines a natural transformation, which we now check. Let H {\displaystyle H} be a group. For any homomorphism f : G → H {\displaystyle f:G\to H} , we have that [ G , G ] {\displaystyle [G,G]} is contained in the kernel of π H ∘ f {\displaystyle \pi _{H}\circ f} , because any homomorphism into ...
The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a linear isomorphism. In the case where =, a linear map is called a linear endomorphism.