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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 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 morphism of schemes is a universal homeomorphism if and only if it is integral, radicial and surjective. [1] In particular, a morphism of locally of finite type is a universal homeomorphism if and only if it is finite, radicial and surjective. For example, an absolute Frobenius morphism is a universal homeomorphism.
This map is a linear injection and for every compact subset (where is also a compact subset of since ) we have ((;)) = (;) (()) If I is restricted to (;) then the following induced linear map is a homeomorphism (and thus a TVS-isomorphism): (;) (;) and thus the next two maps (which like the previous map are defined by ()) are topological ...
In field theory, an embedding of a field in a field is a ring homomorphism:. The kernel of is an ideal of , which cannot be the whole field , because of the condition = =. Furthermore, any field has as ideals only the zero ideal and the whole field itself (because if there is any non-zero field element in an ideal, it is invertible, showing the ...
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.
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.
A homeomorphism is an isomorphism of topological spaces. A diffeomorphism is an isomorphism of spaces equipped with a differential structure, typically differentiable manifolds. A symplectomorphism is an isomorphism of symplectic manifolds. A permutation is an automorphism of a set.