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An action of a topological group G on a topological space X is a group action of G on X such that the corresponding function G × X → X is continuous. Likewise, a representation of a topological group G on a real or complex topological vector space V is a continuous action of G on V such that for each g ∈ G, the map v ↦ gv from V to ...
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 mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group.
In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G→G and the inverse operation G→G are continuous maps. Subcategories This category has the following 2 subcategories, out of 2 total.
If every conjugacy class in a locally compact group has compact closure, then the group is amenable. Examples of groups with this property include compact groups, locally compact abelian groups, and discrete groups with finite conjugacy classes. [9] By the direct limit property above, a group is amenable if all its finitely generated subgroups ...
A profinite group is a topological group that is isomorphic to the inverse limit of an inverse system of discrete finite groups. [3] In this context, an inverse system consists of a directed set (,), an indexed family of finite groups {:}, each having the discrete topology, and a family of homomorphisms {:,,} such that is the identity map on and the collection satisfies the composition ...
Such topological groups are necessarily Hausdorff. Interest centres on locally compact totally disconnected groups (variously referred to as groups of td-type, [1] locally profinite groups, [2] or t.d. groups [3]). The compact case has been heavily studied – these are the profinite groups – but for a long time not much was known about the ...