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