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. 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 X {displaystyle X} be a topological space and denote the homeomorphism group of X {displaystyle X} by G {displaystyle G} . The action is defined as follows: G × X ⟶ X ( φ , x ) ⟼ φ ( x ) {displaystyle {egin{aligned}G imes X&longrightarrow X\(varphi ,x)&longmapsto varphi (x)end{aligned}}} This is a group action since for all φ , ψ ∈ G {displaystyle varphi ,psi in G} , φ ⋅ ( ψ ⋅ x ) = φ ( ψ ( x ) ) = ( φ ∘ ψ ) ( x ) {displaystyle varphi cdot (psi cdot x)=varphi (psi (x))=(varphi circ psi )(x)} where ⋅ {displaystyle cdot } denotes the group action, and the identity element of G {displaystyle G} (which is the identity function on X {displaystyle X} ) 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 H o m e o ( X ) {displaystyle Homeo(X)} becomes a topological group as one can easily show. If X {displaystyle X} 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 H o m e o ( X ) {displaystyle Homeo(X)} therefore not a topological group.