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enter image description here [Vector Bundle Chart lemma]

The above is an example in Lee's smooth manifolds. My question is about the case where $S$ an immersed (or embedded) submanifold of $M$. By the chart lemma, $E|_S$ is indeed a vector bundle. Then is $E|_S$ an immersed (or embedded) submanifold of $E$?

In both cases, immersion of the inclusion map $i:E|_S \rightarrow E$ can be easily checked by coordinate charts in $E|_S$ and $E$. So, how to prove whether the topology of $E|_S$ is the subspace topology of $M$?

gsoldier
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  • $E\vert_S$ is defined as a subspace of $E$ equipped with the subspace topology. So the compatible charts are all you had to check. – Thorgott Feb 21 '23 at 12:27
  • There's a correction to this example in my correction list (https://sites.math.washington.edu/~lee/Books/ISM/). – Jack Lee Feb 21 '23 at 15:45

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