0
$\begingroup$

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?

$\endgroup$

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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

3
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ 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

Not the answer you're looking for? Browse other questions tagged or ask your own question.