 In descriptive set theory, a parallel sequence of topological definitions of subclasses of \(\mathcal{C}\) is given in the context of defining the Borel sets and projective sets. First recall that one means of defining a topology on \(\mathcal{C}\) is to take as basic open sets all sets of functions \(f: \mathbb{N} \rightarrow \{0,1\}\) such that \(\overline{f}(k) = \sigma\) for some \(\sigma \in 2^{&lt; \mathbb{N}}\) and \(k \in \mathbb{N}\). The boldface Borel Hierarchy on \(\mathcal{C}\) is now given by defining \(\mathbf{\Sigma^0_1}\) to be the collection of all open sets of \(\mathcal{C}\), \(\mathbf{\Pi^0_{n}}\) (for \(n \geq 1\)) to be the set of all complements \(\overline{A}\) of sets \(A \in \mathbf{\Sigma^0_1}\), and \(\mathbf{\Sigma^0_{n+1}}\) to be the set of all countable unions \(\bigcup_{i \in \mathbb{N}} A_i\) where \(A_i \in \mathbf{\Pi^0_n}\). (Thus \(\mathbf{\Pi^0_1}\) denotes the set of closed sets, \(\mathbf{\Sigma^0_2}\) denotes the so-called \(F_{\sigma}\) sets, \(\mathbf{\Pi^0_2}\) the \(G_{\delta}\) sets, etc.) The union of these classes corresponds to the boldface Borel sets \(\mathbf{B}\) which may also be characterized as the smallest class of sets containing the open sets of \(\mathcal{C}\) which is closed under countable unions and complementation. The so-called analytic sets are defined to be the continuous images of the Borel sets and are denoted by \(\mathbf{\Sigma^1_1}\) while the co-analytic sets are defined to be the complements of analytic sets and are denoted by \(\mathbf{\Pi^1_1}\). Finally, \(\mathbf{\Delta^1_1}\) is used to denote the intersection of the analytic and co-analytic sets.