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In Proposition 5.47, $M$ is a smooth manifold, but in Theorem 6.15, $M$ is a smooth manifold with or without boundary, how to conclude each $E_i$ is a compact regular domain by Proposition 5.47?

  • $\begingroup$ I believe Proposition 5.47 holds for manifolds with or without boundary. Maybe check the proof of 5.47 (and the theorem it aludes to) to see whether it works for manifolds with boundary as well. $\endgroup$ – quarague Jul 11 at 14:00
  • $\begingroup$ Can you prove it? $\endgroup$ – Born to be proud Jul 11 at 23:36

This excerpt is from my Introduction to Smooth Manifolds (2nd ed.). There's a correction to this proof on my website. Does that answer your question?

  • $\begingroup$ But how to show $E_i$ is a smooth manifold with or without boundary? $\endgroup$ – Born to be proud Jul 12 at 7:22
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    $\begingroup$ It might not be. That's the point of the correction -- the argument works for any compact subset of a smooth manifold with or without boundary, and $E_i$ is such a subset. $\endgroup$ – Jack Lee Jul 12 at 14:39

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