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

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 ...
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What is/are the definitions of local diffeomorphism onto image?

In summary: Actually, I think the confusion arises from a distinction between (local diffeomorphism)-onto image and local-(diffeomorphism onto image). See (C1) at the end. Firstly, I believe this is ...
Is $x^3-6xy+y^2=-108$ a regular submanifold but not a regular $k$-submanifold?
My book is An Introduction to Manifolds by Loring W. Tu. Let $S = \{x^3-6xy+y^2=-108\}$, and let "submanifold" and "$k$-submanifold" mean, respectively, "regular" and "regular $k$-submanifold". As in ...