It is not clear to me if the disjoint union of a $d_{1}$-manifold and a $d_{2}$-manifold is still a manifold?

Question

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?

Randall · Answer 1 · 2019-09-23T01:39:13.913

3

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.

edited Sep 23 '19 at 01:39

answered Sep 23 '19 at 01:24

Randall

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It depends on what you want to take as your definition. – Santana Afton Sep 23 '19 at 01:48
If they are not just topological manifolds but smooth manifolds, what will be the conclusion? – Zhang Feng Sep 26 '19 at 03:26
The same, since invariance of domain applies to homeomorphism types, not just diffeo types. – Randall Sep 26 '19 at 09:25

It is not clear to me if the disjoint union of a $d_{1}$-manifold and a $d_{2}$-manifold is still a manifold?

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