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  "body": "\n## Summary\n\nThe identity component of a [[topological group|92700QAB5Y]] $G$ is the [[connected component|0JG9W3KA6B]] of $G$ that contains the identity element.\nIt is a closed normal subgroup of $G$, and the other connected components are its cosets.\nThe quotient of $G$ by its identity component is totally disconnected.\nSo every topological group is built from a connected group and a totally disconnected group.\n\n## Explanations\n\n### L1 human\n\nStart at the identity of the group and collect every element you can reach without jumping across a gap.\nThe elements you collect form the identity component.\nFor example, in the group of invertible real $2 \\times 2$ matrices, the identity component consists of the matrices with positive determinant.\nThe matrices with negative determinant form the other piece.\nTo reach them from the identity you would have to pass through a matrix of determinant zero, which is not in the group.\n",
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