Is this a manifold with boundary?

Question

I am new to differential geometry/manifolds. I have two questions:

1) Are two disks tangentially touching each other a manifold with boundary. ie: is this set a manifold with boundary

$$\{(x,y):(x-1)^2+y^2\leq 1\} \cup \{(x,y):(x+1)^2+y^2\leq 1 \}$$

2) If this isn't a manifold with boundary, is there some other special term for the above in differential geometry?

Thanks.

Answer 1

This is not a manifold with boundary because the point $(0,0)$ (which you can check is a boundary point) does not have a neighborhood homeomorphic to the half disk $\{(x,y) \mid x^2+y^2 < 1, x\geq 0\}$.