basic results in topological groups
The purpose of this entry is to list some and useful results concerning the topological of topological groups. We will use the following notation whenever A,B are subsets of a topological group G and r an element of G:
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Ar:={ar:a∈A}
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rA:={ra:a∈A}
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AB:={ab:a∈A,b∈B}
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A2:={a1a2:a1,a2∈A}
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A-1:={a-1:a∈A}
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ˉA denotes the closure
of A
1 - Let G be a topological group and r∈G. The left multiplication s↦rs, multiplication s↦sr, and inversion s↦s-1, are homeomorphisms of G.
2 - Let G be a topological group and e∈G the identity element. Let ℬ be a neighborhood base around e. Then {Br}B∈ℬ is a neighborhood base around r∈G and {Br:B∈ℬ and r∈G} is a basis (http://planetmath.org/BasisTopologicalSpace) for the topology
of G.
3 - Let G be a topological group. If U⊆G is open and V is any subset of G, then UV is an open set in G.
4 - Let G be a topological group and K,L compact sets in G. Then KL is also compact.
5 - Let G be a topological group and e∈G the identity element. If V is a neighborhood of e then V⊂ˉV⊂V2.
6 - Let G be a topological group, e∈G the identity element and W a neighborhood around e. Then there exists a neighborhood U around e such that U2⊂W.
7 - Let G be a topological group, e∈G the identity element and W a neighborhood around e. Then there exists a symmetric (http://planetmath.org/SymmetricSet) neighborhood U around e such that U2⊆W.
8 - Let G be a topological group. If H is a subgroup of G, then so is ˉH.
9- Let G be a topological group. If H is an open subgroup of G, then H is also closed.
Title | basic results in topological groups |
---|---|
Canonical name | BasicResultsInTopologicalGroups |
Date of creation | 2013-03-22 17:37:38 |
Last modified on | 2013-03-22 17:37:38 |
Owner | asteroid (17536) |
Last modified by | asteroid (17536) |
Numerical id | 16 |
Author | asteroid (17536) |
Entry type | Result |
Classification | msc 22A05 |
Related topic | PolishGSpace |
Related topic | PolishGroup |