This cute set theory puzzle requires some mathematical maturity.

Does a function $f:[0, 1]\to \mathcal{P}([0, 1])$ with the following properties exist:

1. \[\forall x \in [0, 1]. |f(x)| \leq \aleph_{0}, \text{ i.e. } f(x) \text{ is countable.}\]
2. \[\forall x, y \in [0, 1]. x \in f(y) \text{ or } y \in f(x).\]

How does your answer change if you assume that the continuum hypothesis is true vs if you assume that it is false?