For a topological space \(X\), let \(2^X\) be the set of all closed subsets of \(A\) and \(\text{CL}(X)=2^X-\{\emptyset\}\). \(\Delta\) denotes a subset of \(\text{CL}(X)\) satisfying the following properties: (a) \(\Delta\) is closed with respect to the finite unions; (b) the intersection of every finite subset of \(\Delta\) belongs to \(\Delta\) if it contains an element of \(\Delta\). Let \((X,\tau)\) be a topological space and let \(\Delta\subset\text{CL}(X)\). The hit-and-miss topology \(\tau_\Delta\) on \(2^X\) has as a subbase all sets of the form \(V^-\), where \(V\in\tau\), and of the form \((A^C)^+\), where \(A\in \Delta\), and the space \((2^X, \tau_\Delta)\) is denoted by \(\Delta(X)\); for a subset \(A\) of \(X\), put \(A^-= \{F\in 2^X: F\cap A\neq \emptyset\}\), \(A^+=\{F\in 2^X: F\cap A^C\neq \emptyset\}\), where \(A^C\) denotes the complement of \(A\). Let \((X,\tau)\) be a topological space. For a subset \({\mathcal A}\) of \(\text{CL} (X)\), a subset \({\mathcal B}\) of \({\mathcal A}\) is said to be \(\tau\)-cofinal, if for any \(W\in\tau\) and \(A\in{\mathcal A}\) with \(W\cap A^C\neq \emptyset\), there exists \(B\in{\mathcal B}\) which contains \(A\) and \(W\cap B^C\neq \emptyset\) holds. The \(\tau\)-cofinality \(\tau k({\mathcal A})\) of \({\mathcal A}\) is defined as follows: \(\tau k ({\mathcal A})= \inf\{|{\mathcal B}|:{\mathcal B}\) is a \(\tau\)-cofinal subset of \({\mathcal A}\}+\omega\). The author discusses the cardinal invariant associated with \(\Delta\) and proves as main results the following theorems. Theorem 3.1. Let \((X,\tau)\) be either a quasi-regular and \(R_0\) or a \(T_1\) space. Then \[ \pi w(\Delta(X))= \pi\chi (\Delta(X))= \max\{\pi w(X), \tau k(\Delta)\}, \] where \(\pi w\) and \(\pi\chi\) denote the \(\pi\)-weight and \(\pi\)-character, respectively. Theorem 3.5. Let \(X\) be an \(R_0\) space. Then \(\pi w((2^X, \tau_v))= \pi w(X)\), where \(\tau_v\) denotes \(\tau_\Delta\) for \(\Delta= \text{CL} (X)\). A topological space \(X\) is said to be quasi-regular whenever each nonempty open subset of \(X\) contains a closed subset whose interior is nonempty, and \(X\) is an \(R_0\) space whenever each nonempty open subset of \(X\) contains the closure of each of its points.