On the index sets of $\Sigma$-subsets of the real numbers

We compute the levels of complexity in analytical and arithmetical hierarchies for the sets of the $\Sigma$-formulas defining in the hereditarily finite superstructure over the ordered field of the reals the classes of open, closed, clopen, nowhere dense, dense subsets of $\mathbb R^n$, first category subsets in $\mathbb R^n$ as well as the sets of pairs of $\Sigma$-formulas corresponding to the relations of set equality and inclusion which are defined by them. It is also shown that the complexity of the set of the $\Sigma$-formulas defining connected sets is at least $\Pi^1_1$.