Explore: Topological Properties Of Groups Of Homeomorphisms Or Diffeomorphisms

function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that

examples of topological groups that are isomorphic as ordinary groups but not as topological groups. Indeed, any non-discrete topological group is also

given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and topological groups. More precisely, a homogeneous

compactness of the quotient space X / G. Now assume G is a topological group and X a topological space on which it acts by homeomorphisms. The action

homeomorphism, diffeomorphism, isometry) between topological spaces, two spaces are said to be locally equivalent if every point of the first space has

self-homeomorphisms of the torus (SL mapping to orientation-preserving maps), and in fact map isomorphically to the (extended) mapping class group of the

linear groups or matrix groups (the automorphism group GL ⁡ ( V ) {\displaystyle \operatorname {GL} (V)} is a linear group but not a matrix group). These

about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to

that conformal homeomorphisms on the Riemann sphere are exactly the Möbius transformations, which can further be identified as elements of the projective

the topological group of diffeomorphisms of euclidean space R n {\displaystyle \mathbb {R} ^{n}} . An inductive limit yields topological groups Top {\displaystyle