Pinning down versus density (Q501845)

\textit{T. Banakh} and \textit{A. Ravsky} [Topology Appl. 201, 181--205 (2016; Zbl 1338.54032)] introduced the foredensity of a topological space \((X,\tau)\) as the smallest cardinal \(\kappa\) such that for any neighbourhood assignment \(U:X\to\tau\) there is \(A\subset X\) of cardinality at most \(\kappa\) with \(A\cap U(x)\not=\varnothing\) for all \(x\). In the present paper this is renamed the pinning down number and denoted by pd\((X)\). It is shown that pd\((X)=\text{d}(X)\) for each (0-dimensional) Hausdorff space \(X\) if and only if \(2^\kappa<\kappa^{+\omega}\) for each cardinal \(\kappa\). It is shown that pd\((X)=\text{d}(X)\) for any locally compact Hausdorff space. Equiconsistency between Shelah's Strong Hypothesis and conditions involving pd\((X)<\text{d}(X)\) are also exhibited.