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[parent] uniform structure of a topological group (Derivation)

Let $ G$ be a topological group. There is a natural uniform structure on $ G$ which induces its topology. We define a subset $ V$ of the Cartesian product $ G \times G$ to be an entourage if and only if it contains a subset of the form

$\displaystyle V_N = \{ (x,y) \in G \times G : xy^{-1} \in N \} $
for some $ N$ neighborhood of the identity element. This is called the right uniformity of the topological group, with which right multiplication becomes a uniformly continuous map. The left uniformity is defined in a similar fashion, but in general they don't coincide, although they both induce the same topology on $ G$.



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"uniform structure of a topological group" is owned by mps. [ full author list (3) | owner history (1) ]
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Also defines:  right uniformity, left uniformity

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Cross-references: map, uniformly continuous, multiplication, identity element, neighborhood, contains, entourage, Cartesian product, subset, topology, induces, uniform structure, topological group
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This is version 7 of uniform structure of a topological group, born on 2002-06-13, modified 2007-06-28.
Object id is 3104, canonical name is UniformStructureOfATopologicalGroup.
Accessed 2675 times total.

Classification:
AMS MSC: 54E15 (General topology :: Spaces with richer structures :: Uniform structures and generalizations)
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