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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 ...
Topological space; Topological property; Open set, closed set. Clopen set; Closure (topology) Boundary (topology) Dense (topology) G-delta set, F-sigma set; closeness (mathematics) neighbourhood (mathematics) Continuity (topology) Homeomorphism; Local homeomorphism; Open and closed maps; Germ (mathematics) Base (topology), subbase; Open cover ...
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 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.
The mapping class group can be defined for arbitrary manifolds (indeed, for arbitrary topological spaces) but the 2-dimensional setting is the most studied in group theory. The mapping class group of surfaces are related to various other groups, in particular braid groups and outer automorphism groups .
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 algebraic topology, the fundamental group (,) of a pointed topological space (,) is defined as the group of homotopy classes of loops based at .This definition works well for spaces such as real and complex manifolds, but gives undesirable results for an algebraic variety with the Zariski topology.