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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 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 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)} .
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 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.
In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation.
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.