Identity component

In mathematics, the identity component of a topological group G is the connected component G₀ of G that contains the identity element of the group. Similarly, the identity path component of a topological group G is the path component of G that contains the identity element of the group.

The identity component G₀ of a topological group G is a closed normal subgroup of G. It is closed since components are always closed. It is a subgroup since multiplication and inversion in a topological group are continuous maps by definition. Moreover, for any continuous automorphism a of G we have

Thus, G₀ is a characteristic subgroup of G, so it is normal.

The identity component G₀ of a topological group G need not be open in G. In fact, we may have G₀ = {e}, in which case G is totally disconnected. However, the identity component of a locally path-connected space (for instance a Lie group) is always open, since it contains a path-connected neighbourhood of {e}; and therefore is a clopen set.

The identity path component may in general be smaller than the identity component (since path connectedness is a stronger condition than connectedness), but these agree if G is locally path-connected.

The quotient group G/G₀ is called the group of components or component group of G. Its elements are just the connected components of G. The component group G/G₀ is a discrete group if and only if G₀ is open. If G is an affine algebraic group then G/G₀ is actually a finite group.

...
Wikipedia