Topological group

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The article on this topic in the Group Properties Wiki can be found at: topological group

Definition

Symbol-free definition

A topological group is a set endowed with the following two structures:

  • The structure of a group, viz a binary operation called multiplication or product, a unary operation called the inverse map, and a constant called the identity element satisfying the conditions for a group
  • The structure of a topological space

such that the following compatibility conditions are satisfied:

  • The inverse map is a continuous map from the group to itself (as a topological space map)
  • The group multiplication map is a jointly continuous map i.e. a continuous map from the Cartesian product of the group with itself, to the group (where the Cartesian product is given the product topology).

Some people assume a topological group to be T_0, that is, that there is no pair of points with each in the closure of the other. This is not a very restrictive assumption, because if we quotient out a topological group by the closure of the identity element, we do get a T_0-topological group. Further information: T0 topological group

Definition with symbols

A topological group is a set G endowed with two structures:

  • The structure of a group viz a multiplication * and an inverse map g \mapsto g^{-1} and an identtiy element e.
  • The structure of a topological space viz a topology \tau

such that:

Relation with other structures

Weaker structures

Facts

Not every topological space can be realized as the underlying space of a topological group. If we restrict attention to T0 topological groups, then the underlying space of a T0 topological group must be completely regular and homogeneous. There are other conditions that need to be satisfied. On the other hand, not every topological group is metrizable, or even normal.

Further information: Topological group not implies normal