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A topological group, G, is a topological space that is also a group such that the group operation (in this case product): ⋅ : G × G → G, (x, y) ↦ xy. and the inversion map: −1 : G → G, x ↦ x−1. are continuous. [note 1] Here G × G is viewed as a topological space with the product topology. Such a topology is said to be compatible ...
Category. : Topological groups. Wikimedia Commons has media related to Topological groups. 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.
Fundamental group. 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.
A torus, one of the most frequently studied objects in algebraic topology. Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
Compact group. The circle of center 0 and radius 1 in the complex plane is a compact Lie group with complex multiplication. In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group).
For example, if two topological objects have different homotopy groups, they cannot have the same topological structure—a fact that may be difficult to prove using only topological means. For example, the torus is different from the sphere: the torus has a "hole"; the sphere doesn't. However, since continuity (the basic notion of topology ...
In group theory, a complete topological group is one which is complete with respect to both of its uniformities. Category: Topological groups.
The completion of a balanced group with respect to its uniform structure admits a unique topological group structure extending that of .This generalizes the case of abelian groups and is a special case of the two-sided completion of an arbitrary topological group, which is with respect to the coarsest uniform structure finer than both the left and the right uniform structures.