Power set

The power set of a set $A$ is

Power sets organize all possible subcollections of $A$. In analysis they occur implicitly whenever one studies families of subsets (e.g., open sets, measurable sets) as subsets of $\mathcal{P}(X)$ for some ambient set $X$.

Examples:

• If $A=\{1,2\}$, then $\mathcal{P}(A)=\{\varnothing,\{1\},\{2\},\{1,2\}\}$.
• $\mathcal{P}(\varnothing)=\{\varnothing\}$.
• If $A$ is finite with $|A|=n$, then $|\mathcal{P}(A)|=2^n$.