Is it true that , in a Euclidean space , union of two disjoint smooth manifolds of different dimensions can never be a manifold?

Question

Let $A,B \subseteq \mathbb R^k$ be disjoint smooth manifolds of different dimensions , then is it rue that $A \cup B $ cannot be a smooth manifold ?

Answer 1

In ${\mathbb R}^2$ let $$A:=\bigl\{(x,y)\,\bigm|\,1<x^2+y^2<4\bigr\}\setminus\bigl\{(x,0)\,\bigm|\,1<x<2\bigr\}\ ,\quad B:=\bigl\{(x,0)\,\bigm|\,1<x<2\bigr\}\ .$$