Connected and Path Connected

Quote:

Which of the following subsets of

are a). path connected b). connected:

i).

ii).

iii).

I think that:

i). I don't think this is path connected. If you took a point in each set , for example and , 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!