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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.
A continuous map between two topological spaces induces a group homomorphism between the associated homotopy groups. In particular, if the map is a continuous bijection (a homeomorphism ), so that the two spaces have the same topology, then their i -th homotopy groups are isomorphic for all i .
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.
A group homomorphism between topological groups is continuous if and only if it is continuous at some point. [4] An isomorphism of topological groups is a group isomorphism that is also a homeomorphism of the underlying topological spaces. This is stronger than simply requiring a continuous group isomorphism—the inverse must also be continuous.
Homotopy groups are such a way of associating groups to topological spaces. A torus A sphere. That link between topology and groups lets mathematicians apply insights from group theory to topology. For example, if two topological objects have different homotopy groups, they cannot have the same topological structure—a fact that may be ...
In the algebraic approach, one finds a correspondence between spaces and groups that respects the relation of homeomorphism (or more general homotopy) of spaces. This allows one to recast statements about topological spaces into statements about groups, which have a great deal of manageable structure, often making these statements easier to prove.
A space X is (topologically) homogeneous if for every x and y in X there is a homeomorphism : such that () =. Intuitively speaking, this means that the space looks the same at every point. All topological groups are homogeneous. Finitely generated or Alexandrov.
As with other sets of maps between topological spaces, the homeomorphism group can be given a topology, such as the compact-open topology.In the case of regular, locally compact spaces the group multiplication is then continuous.