I'm stuck on this (apparently) simple thing:
If $G$ is a topological group and $G_0$ is the connected component of $G$ containing the identity then $G/G_0$ is discrete if and only if $G_0$ is open.
Can you help me to figure out why? Thank you
2026-04-01 16:00:11.1775059211
Quotient group $G/G_0$ in Group Topology
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The quotient group $G/G_0$ is discrete if and only if singleton sets are open, which is the case if and only if the unit element singleton is open. The preimage of the unit element singleton of $G/G_0$ under the canonical projection is $G_0$.