# Union of submanifolds

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.

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