| 1. | The theorem also has consequences for broader classes of topological groups.
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| 2. | The classical groups are important examples of non-abelian topological groups.
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| 3. | This much is a fragment of a typical locally Euclidean topological group.
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| 4. | A relatively refined theory is available for pseudocompact topological groups.
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| 5. | More generally, every commutative topological group is also a uniform space.
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| 6. | Let G be a compact topological group, with a distance d.
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| 7. | For topological groups, the quotient map is open.
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| 8. | Thus, the idele group is a topological group.
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| 9. | Therefore, the group of units is not a topological group in general.
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| 10. | Additionally, the absolute value function induces the following isomorphisms of topological groups:
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