A generalization of the equality \(p(X)=a(C_p (X))\) (Q2467006)

Let \(X\) be an infinite Tikhonov space and \(T(X)\) the set of all nonempty open subsets of \(X\). The author defines cardinal invariants \(p^\alpha_m(X)\) and \(\alpha_m(X)\) which are generalizations of the point-finite cellularity \(p(X)\) and the Alexandroff number \(\alpha(X)\), as below: For two families \(\alpha\) and \(\beta\) of subsets of \(X\), if for every \(A\in\alpha\), \(\text{card}(\{B\in\beta: A\cup\neq B\})< m\), then \(\beta\) is said to be \(\alpha-< m\). For a family \(\alpha\) of subets of \(X\) and a cardinal \(\tau\), if there is an \(\alpha-< m\) family \({\mathcal V}\subset T(X)\) such that \(\text{card}({\mathcal V})= \tau\), then it is denoted by \(X\in P^\alpha_m(\tau)\). Let \(A\) be a discrete space of cardinality \(\tau\), and let \(\Omega\) be an object not in \(A\). Then \(A_m(\tau)\) denotes the set of \(A\cup\{\Omega\}\) with following topology: open sets in \(A_m(\tau)\) are sets of the form \(\{\Omega\}\cup(A\setminus F)\), where \(F\subset A\) and \(|F|< m\), together with all subsets of \(A\). A class \(Q_m(\tau)\) of spaces is as follows: \(Y\in Q_m(\tau)\) if and only if there is a continuous one-to-one mapping \(\varphi: A_m(\tau)\to Y\). The generalized Alexandroff number \(\alpha(Y)\) of \(Y\) is defined to be the supremum of the cardinal \(\tau\) such that \(Y\in Q_m(\tau)\). The author proves the following result, which generalizes Tkachuk's equality \(p(X)= \alpha(C_p(X))\), in the case \(m=\aleph_0\). Theorem (2.3). For any cover \(\alpha\) of \(X\), the equality \(m\cdot p^\alpha_m(X)= m\cdot\alpha_m(C_\alpha(X))\) holds. For a family \(\alpha\) of subsets of \(X\), \(C_\alpha\) is the function space with the topology on \(C(X)\) induced by \(\{[A, V]: A\in\alpha\) and \(V\) is open in \(\mathbb{R}\}\) as subbase, where \([A, V]= \{f\in C(X): f(A)\subset V\}\).