| 21. | This need not be true in a general abelian topological group ( see examples below ).
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| 22. | A topological group is a mathematical object with both an algebraic structure and a topological structure.
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| 23. | Topological groups, along with continuous group actions, are used to study continuous in physics.
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| 24. | Every subgroup of a topological group is itself a topological group when given the subspace topology.
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| 25. | Every subgroup of a topological group is itself a topological group when given the subspace topology.
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| 26. | For example, a topological group is just a group in the category of topological spaces.
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| 27. | In 1934, Lev Pontryagin proved the Pontryagin duality theorem; a result on topological groups.
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| 28. | The Bredon cohomology of topological spaces under action of a topological group is named after him.
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| 29. | In any case, any topological group can be made Hausdorff by taking an appropriate canonical quotient.
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| 30. | Partly for this reason, it is natural to concentrate on closed subgroups when studying topological groups.
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