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Let $M$ be a smooth manifold (without boundary) and $A,B$ too submanifolds of $M$ such that $$A\cap B=\emptyset\quad\text{and}\quad\dim A=\dim B.$$ Is $A\cup B$ a submanifold of $M$?

The assumption that $\dim A=\dim B$ is really necessary. For example, $A=\{0\}$ and $B=(0,1)$ in $\Bbb R$ are disjoint submanifolds, but $A\cup B=[0,1)$ is not a submanifold.

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Not necessarily. For example, in $\Bbb R^2$, take $$ \begin{align} A &= \{(0,y):y\in\Bbb R\}\quad\text{(the $y$-axis)}\\ B &= \{(x,0):x>0\}\quad\text{(the positive $x$-axis)}. \end{align} $$ Their union is not a manifold near $(0,0)$; it has a $\vdash$ shape.

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No it's not true in general. Consider for instance two lines in $\mathbb{R}^2$, their union is not a submanifold.

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