A Borel set of $\mathbb R$ is equivalent to an interval. An element of $\mathscr P(\mathbb R)$ is also an interval.
So, can we say a Borel set on $\mathbb R$ is a part of $\mathbb R$?
More generally, what is the use of Borel sets of a set $X$ if we already have $\mathscr P(X)$ there for us?
PS: I am not very confident with Borel sets. What I am stating above might be wrong...
A Borel set is absolutely a part of $\mathbb{R}$ as it is generated by the open sets (as well as the closed and compact sets) on $\mathbb{R}$. Any set that you can imagine or write down is pretty much guaranteed to be a Borel Set. Singletons? Borel. Closed sets? Borel. Half open to infinity? Borel.
Why do we need Borel sets and not just the power set? It is due to the existence of some strange sets on $\mathcal{P}(\mathbb{R})$ that aren't measureable/can't be assigned a meaningful length, although concretely writing such a set on a piece of paper is impossible. But they do exist. For further reading on this matter, check out Vitali's set.