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If $Y$ is a subspace of $X$, and $C$ is a connected component of $Y$, then C need not be a connected component of $X$ (take for instance two disjoint open discs in $\mathbb{R}^2$).

But I read that, under the same hypothesis, $C$ need not even be connected in $X$. Could you please provide me with an example, or point me towards one?

Thank you.

SOURCE http://www.filedropper.com/manifolds2 Page 129, paragraph following formula (A.7.16).

  • 0
    What do you mean by 'connected in $X$'? Connected viewed as a subspace of $X$?2011-10-07
  • 3
    connectedness is like compactness an intrinsic property: it does not matter how a subspace sits in a space.2011-10-07

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