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In mathematics, the identity component of a topological group ''G'' is the connected component ''G''0 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. == Properties == The identity component ''G''0 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''(''G''0) = ''G''0. Thus, ''G''0 is a characteristic subgroup of ''G'', so it is normal. The identity component ''G''0 of a topological group ''G'' need not be open in ''G''. In fact, we may have ''G''0 = , 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 ; 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Identity component」の詳細全文を読む スポンサード リンク
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