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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 compactly generated (topological) group is a topological group G which is algebraically generated by one of its compact subsets. [1] This should not be confused with the unrelated notion (widely used in algebraic topology) of a compactly generated space-- one whose topology is generated (in a suitable sense) by its compact subspaces.
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).
If : is a continuous group homomorphism of topological groups and if X is a G-space, then H can act on X by restriction: = (), making X a H-space. Often f is either an inclusion or a quotient map. In particular, any topological space may be thought of as a G -space via G → 1 {\displaystyle G\to 1} (and G would act trivially.)
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.
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 semitopological group is a topological space with a group action that is continuous with respect to each variable considered separately. It is a weakening of the concept of a topological group ; all topological groups are semitopological groups but the converse does not hold.
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.