Suppose that we have the finite set $S^1=\{a,b,c\}$. Easily we can find the power set $\mathcal{P}(S^1)$ of the set, that has $2^{\mid S^1\mid}=2^3=8$ elements where $\mid S^1 \mid$ denotes the cardinal of a set. Then the powerset is $\mathcal{P}(S^1)=\left\{\emptyset, \{a\},\{b\},\{c\}, \{a,b\},\{a,c\},\{b,c\}, S^1\right\}$. I have the following questions:
Is the powerset related to the Borelian set of $S^1$, let me denote this as $\mathcal{B}(S^1)$? I thought that in the finite sets the powerset and the Borelian sets coincide, if not what is thier relation?
Is there any difference in the properties between the powerset and the Borelian set?
Is there any example of how to construct the Borelian set of a subset of $\mathbb{R}^k$ in the forum of MSE?