Definition:Power Set

The power set of a set $$S$$, denoted $$\mathcal{P} \left({S}\right)$$, is the set defined as follows:

$$\mathcal{P} \left({S}\right) = \left\{ {T: T \subseteq S}\right\}$$

That is, the set whose elements are all of the subsets of $$S$$.

Note that this is a set all of whose elements are themselves sets.

Empty Set
It's worth just confirming the obvious:

$$\forall S: \varnothing \in \mathcal{P} \left({S}\right)$$

Proof:

$$\forall S: \varnothing \subseteq S$$ Empty Set Subset of All

$$\Longrightarrow \varnothing \in \mathcal{P} \left({S}\right)$$ (Definition of Power Set)