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A homeomorphism is sometimes called a bicontinuous function. If such a function exists, and are homeomorphic. A self-homeomorphism is a homeomorphism from a topological space onto itself. Being "homeomorphic" is an equivalence relation on topological spaces. Its equivalence classes are called homeomorphism classes.
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".
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 ...
An example is the bipartite double cover, formed from a graph by splitting each vertex v into v 0 and v 1 and replacing each edge u,v with edges u 0,v 1 and v 0,u 1. The function mapping v 0 and v 1 in the cover to v in the original graph is a homomorphism and a covering map. Graph homeomorphism is a different notion, not related directly to ...
Let M be a topological space.A chart (U, φ) on M consists of an open subset U of M, and a homeomorphism φ from U to an open subset of some Euclidean space R n.Somewhat informally, one may refer to a chart φ : U → R n, meaning that the image of φ is an open subset of R n, and that φ is a homeomorphism onto its image; in the usage of some authors, this may instead mean that φ : U → R n ...
In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, or the first isomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism. The homomorphism theorem is used to prove the isomorphism theorems.
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.
That is, the maps on X coming from elements of G preserve the structure associated with the category (for example, if X is an object in Diff then the action is required to be by diffeomorphisms). A homogeneous space is a G-space on which G acts transitively. If X is an object of the category C, then the structure of a G-space is a homomorphism: