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?

Answer 1

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.