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Consider the following assertions:

1. $\left\{(x, y) \in \mathbb{R}^{2} \mid x y=1\right\}$ is connected.
2. $\left\{(x, y) \in \mathbb{C}^{2} \mid x y=1\right\}$ is connected.

Which of the following sentences is true?

1. Both (I) and (II) are true.
2. (I) is true but (II) is false.
3. (I) is false but (II) is true.
4. Both (I) and (II) are false.