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Consider the following assertions:

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

Which of the following sentences is true?

- Both (I) and (II) are true.
- (I) is true but (II) is false.
- (I) is false but (II) is true.
- Both (I) and (II) are false.