Does this statement in the picture tells me that "A set is a manifold iff a subset of it is a manifold.(p.28GP)" :

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Could anyone explain this for me please?

  • $\begingroup$ Euclidean open balls are manifolds. A manifold is basically a space which looks locally like a ball (every point has such neighborhood). So, a space is manifold iff it is locally a manifold. $\endgroup$ – Berci Oct 23 '18 at 14:59
  • $\begingroup$ So my interpretation was wrong?@Berci $\endgroup$ – Idonotknow Oct 23 '18 at 15:00
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    $\begingroup$ I would say that that interpretation is wrong. To me, the laconic way to summarize that statement is "A space is a manifold iff it admits an open cover by manifolds" (together with knowledge of what the subspace topology is). $\endgroup$ – K B Dave Oct 23 '18 at 15:07
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    $\begingroup$ You missed a quantifier. A set is a manifold if and only if every subset of a certain kind is a manifold, namely, a certain neighborhood of every point. $\endgroup$ – Lee Mosher Oct 23 '18 at 16:02

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