I'm trying to solve a question which asks me to let $B_1(p)$ denote the unit ball around $p$ in $\mathbb{R}^2$. I'm supposed to decide whether $\overline{B_1((1,0))}\cup B_1((-1,0))$ is connected and/or path connected?
On the one hand I feel like it should be disconnected because I can't see how you can get two non-empty disjoint open sets whose union is the whole set, but on the other hand I think it should be path connected since you could go from any point in $\overline{B_1((1,0))}$ to $(0,0)$ and then presumably into $B_1((-1,0))$ continuously because it gets arbitrarily close to $(0,0)$, and path-connectedness implies connectedness.
I'd appreciate it if you could tell me where I'm going wrong.
Your second thought is correct: that set is is path-connected because, given any two points $p$ and $q$ in it, you can go from $p$ to $(0,0)$ in a straight line and then from $(0,0)$ to $q$, again in a straight line, without leaving your set.
And, since it is path-connected, it is connected.