I think that:Which of the following subsets of $\displaystyle \mathbb{R}^2$ are a). path connected b). connected:

i). $\displaystyle B_1((1,0)) \cup B_1((-1,0))$

ii). $\displaystyle \overline{B_1((1,0))} \cup \overline{B((-1,0))}$

iii). $\displaystyle \overline{B_1((1,0))} \cup B_1((-1,0))$

i). I don't think this is path connected. If you took a point in each set , for example $\displaystyle \frac{1}{2}$ and $\displaystyle -\frac{1}{2}$, you couldn't get from one to the other in a continuous path. This is because you would have to travel through 0, but 0 isn't in either set.

ii). I think this is path connected. Since both sets now contain 0, you could travel between them.

Since they are path connected, the subsets are connected.

iii). I think this is path connected. One set contains 0, the other contains points arbitrarily close to 0. Therefore you could make a continuous map between the two.

Once again, since they are path connected they are connected.

P.S: Thanks to everyone who has helped me with any topology questions over the past few days. You've helped me so much!