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Homeomorphism group

In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation. Homeomorphism groups are very important in the theory of topological spaces and in general are examples of automorphism groups. Homeomorphism groups are topological invariants in the sense that the homeomorphism groups of homeomorphic topological spaces are isomorphic as groups. There is a natural group action of the homeomorphism group of a space on that space. Let be a topological space and denote the homeomorphism group of by . The action is defined as follows: This is a group action since for all , where denotes the group action, and the identity element of (which is the identity function on ) sends points to themselves. If this action is transitive, then the space is said to be homogeneous. 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. If the space is compact and Hausdorff, the inversion is continuous as well and becomes a topological group. If is Hausdorff, locally compact and locally connected this holds as well. However there are locally compact separable metric spaces for which the inversion map is not continuous and therefore not a topological group. In the category of topological spaces with homeomorphisms, group objects are exactly homeomorphism groups. Mapping class group In geometric topology especially, one considers the quotient group obtained by quotienting out by isotopy, called the mapping class group: The MCG can also be interpreted as the 0th homotopy group, . This yields the short exact sequence: In some applications, particularly surfaces, the homeomorphism group is studied via this short exact sequence, and by first studying the mapping class group and group of isotopically trivial homeomorphisms, and then (at times) the extension.

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Space (mathematics)
In mathematics, a space is a set (sometimes called a universe) with some added structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself. A space consists of selected mathematical objects that are treated as points, and selected relationships between these points. The nature of the points can vary widely: for example, the points can be elements of a set, functions on another space, or subspaces of another space.
Homogeneous space
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. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and topological groups. More precisely, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively. The elements of G are called the symmetries of X.
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. To define the n-th homotopy group, the base-point-preserving maps from an n-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes.
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