Properties
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
- a(G0) = G0.
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.
Read more about this topic: Identity Component
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