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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 ...
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.
In mathematics, a topological abelian group, or TAG, is a topological group that is also an abelian group. That is, a TAG is both a group and a topological space, the group operations are continuous, and the group's binary operation is commutative. The theory of topological groups applies also to TAGs, but more can be done with TAGs.
In mathematics, a locally compact group is a topological group G for which the underlying topology is locally compact and Hausdorff.Locally compact groups are important because many examples of groups that arise throughout mathematics are locally compact and such groups have a natural measure called the Haar measure.
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).
A topological group is called locally compact if the underlying topological space is locally compact and Hausdorff; the topological group is called abelian if the underlying group is abelian. Examples of locally compact abelian groups include: for n a positive integer, with vector addition as group operation.
This set (with the group structure described below) is called the fundamental group of the topological space X at the base point . The purpose of considering the equivalence classes of loops up to homotopy, as opposed to the set of all loops (the so-called loop space of X ) is that the latter, while being useful for various purposes, is a ...
The nth homotopy group of a topological space is the group of homotopy classes of basepoint-preserving maps from the -sphere to , under the group operation of concatenation. The most fundamental homotopy group is the fundamental group π 1 ( X ) {\displaystyle \pi _{1}(X)} .