While it is clear that a disjoint union of two $d$-manifolds is a $d$-manifold, it is not clear to me if the disjoint union of a $d_{1}$-manifold and a $d_{2}$-manifold is still a manifold and if yes under some conditions then what is its dimension?


marked as duplicate by Santana Afton, Yanior Weg, Maximilian Janisch, Don Thousand, Shailesh Oct 5 at 2:48

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This is true if and only if $d_1=d_2$. The definition of a manifold mentions only one dimension, which must be consistent across all charts.

I suppose that in order to truly believe this you must also believe in Invariance of Domain.

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    $\begingroup$ It depends on what you want to take as your definition. $\endgroup$ – Santana Afton Sep 23 at 1:48
  • $\begingroup$ If they are not just topological manifolds but smooth manifolds, what will be the conclusion? $\endgroup$ – Zhang Feng Sep 26 at 3:26
  • $\begingroup$ The same, since invariance of domain applies to homeomorphism types, not just diffeo types. $\endgroup$ – Randall Sep 26 at 9:25

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