I know the union of two manifolds, let’s say of dimension k both, is a manifold iff in every point p of the intersection, you can find an open neighborhood in the union such that it looks like (an open subset of) R^{k}. But, how would this work for two manifolds M and N of dimensions m and n respectively? M and N are not disjoint necessarily.
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1Can you state your definition of manifold? Right now it seems that you are considering submanifolds of Euclidean spaces. – Arctic Char Mar 05 '23 at 21:36
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https://math.stackexchange.com/questions/2273320/can-a-topological-manifold-be-non-connected-and-each-component-with-different-di/2273365#2273365 and https://math.stackexchange.com/questions/4637062/sufficient-condition-for-the-union-of-submanifolds-to-be-a-submanifold/4642619#4642619 – Moishe Kohan Mar 05 '23 at 21:41