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In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other. Topological groups have been studied extensively in the period of 1925 to 1940. Haar and Weil (respectively in 1933 and 1940) showed that the integrals and Fourier series are special cases of a very wide class of topological groups. Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics. In functional analysis, every topological vector space is an additive topological group with the additional property that scalar multiplication is continuous; consequently, many results from the theory of topological groups can be applied to functional analysis. A topological group, G, is a topological space that is also a group such that the group operation (in this case product): ⋅ : G × G → G, (x, y) ↦ xy and the inversion map: −1 : G → G, x ↦ x−1 are continuous. Here G × G is viewed as a topological space with the product topology. Such a topology is said to be compatible with the group operations and is called a group topology. Checking continuity The product map is continuous if and only if for any x, y ∈ G and any neighborhood W of xy in G, there exist neighborhoods U of x and V of y in G such that U ⋅ V ⊆ W, where U ⋅ V := {u ⋅ v : u ∈ U, v ∈ V}. The inversion map is continuous if and only if for any x ∈ G and any neighborhood V of x−1 in G, there exists a neighborhood U of x in G such that U−1 ⊆ V, where U−1 := { u−1 : u ∈ U }. To show that a topology is compatible with the group operations, it suffices to check that the map G × G → G, (x, y) ↦ xy−1 is continuous. Explicitly, this means that for any x, y ∈ G and any neighborhood W in G of xy−1, there exist neighborhoods U of x and V of y in G such that U ⋅ (V−1) ⊆ W.

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