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I was reading a book where for $G$ a connected lie group, and $H$ a subgroup of $G$. It proved that $H$ be closed sub lie group of $G$, and then concluded that $H=G$.

Is it correct? If so, why?(Is it because the manifold structure of $G$?)

ShoutOutAndCalculate
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    You can only conclude that if they have the same dimension. – Matt Samuel Oct 09 '19 at 18:52
  • @MattSamuel If they have the same dimension, then it followed from "geometry" that "any point" in $H$ can travel to "anywhere" because G is a connected lie group thus a connected manifold right? – ShoutOutAndCalculate Oct 09 '19 at 18:55
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    No, it's the theorem on invariance of domain. If they have the same dimension, then the image is open, and since it is also closed it must be the whole manifold since it is connected. – Matt Samuel Oct 09 '19 at 18:58

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