Watch out! Your conclusions are correct, but your reasoning is insufficient.

For (1), think carefully about the definition of connectedness.

You (should) know that balls are connected, so (2) is two connected spaces with nonempty intersection, and (3) is two connected spaces each of which intersects a third connected space (think about the path you'd like to draw). There are theorems you can cite.

Edit:I wasn't as clear as I could have been on (3). The reason this one isn't as straightforward as (2) is that the two balls have empty intersection. But onsider the set , which is a straight-line path from the center of one ball to the center of the other. If you call the two balls , then you can write , andthenapply the theorem about connected spaces with nonempty intersection.