Topological Group - Properties

Properties

The algebraic and topological structures of a topological group interact in non-trivial ways. For example, in any topological group the identity component (i.e. the connected component containing the identity element) is a closed normal subgroup. This is because if C is the identity component, a*C is the component of G (the group) containing a. In fact, the collection of all left cosets (or right cosets) of C in G is equal to the collection of all components of G. Therefore, the quotient topology induced by the quotient map from G to G/C is totally disconnected.

The inversion operation on a topological group G is a homeomorphism from G to itself. Likewise, if a is any element of G, then left or right multiplication by a yields a homeomorphism G → G.

Every topological group can be viewed as a uniform space in two ways; the left uniformity turns all left multiplications into uniformly continuous maps while the right uniformity turns all right multiplications into uniformly continuous maps. If G is not abelian, then these two need not coincide. The uniform structures allow one to talk about notions such as completeness, uniform continuity and uniform convergence on topological groups.

As a uniform space, every topological group is completely regular. It follows that if a topological group is T₀ (Kolmogorov) then it is already T₂ (Hausdorff), even T_3½ (Tychonoff).

Every subgroup of a topological group is itself a topological group when given the subspace topology. If H is a subgroup of G, the set of left or right cosets G/H is a topological space when given the quotient topology (the finest topology on G/H which makes the natural projection q : G → G/H continuous). One can show that the quotient map q : G → G/H is always open.

Every open subgroup H is also closed, since the complement of H is the open set given by the union of open sets gH for g in G \ H.

If H is a normal subgroup of G, then the factor group, G/H becomes a topological group when given the quotient topology. However, if H is not closed in the topology of G, then G/H will not be T₀ even if G is. It is therefore natural to restrict oneself to the category of T₀ topological groups, and restrict the definition of normal to normal and closed.

The isomorphism theorems known from ordinary group theory are not always true in the topological setting. This is because a bijective homomorphism need not be an isomorphism of topological groups. The theorems are valid if one places certain restrictions on the maps involved. For example, the first isomorphism theorem states that if f : G → H is a homomorphism then G/ker(f) is isomorphic to im(f) if and only if the map f is open onto its image.

If H is a subgroup of G then the closure of H is also a subgroup. Likewise, if H is a normal subgroup, the closure of H is normal.

A topological group G is Hausdorff if and only if the trivial one-element subgroup is closed in G. If G is not Hausdorff then one can obtain a Hausdorff group by passing to the quotient space G/K where K is the closure of the identity. This is equivalent to taking the Kolmogorov quotient of G.

The fundamental group of a topological group is always abelian. This is a special case of the fact that the fundamental group of an H-space is abelian, since topological groups are H-spaces.