Dimension of k-leaders (Q1819410)

Let us recall for any Hausdorff space \({\mathcal X}\) the notion \({\mathcal K}\)- leader \({\mathcal K}{\mathcal X}\) of \({\mathcal X}\). This is a topological space \({\mathcal K}{\mathcal X}\) with the same objects as \({\mathcal X}\) but the topology in \({\mathcal K}{\mathcal X}\) is defined in this way: a set \({\mathcal F}\) is closed in \({\mathcal K}{\mathcal X}\) iff any intersection of \({\mathcal F}\) with any compact subset of \({\mathcal X}\) is closed in \({\mathcal X}\). The author constructs the following example: there exists a Lindelöf space \({\mathcal X}\) such that \(\dim{\mathcal X}>0\) and every compact subspace of \({\mathcal X}\) is finite. Let us note that this space has the property \(\dim{\mathcal X}>\dim {\mathcal K}{\mathcal X}\), because clearly \({\mathcal K}{\mathcal X}\) is discrete and hence dim \({\mathcal K}{\mathcal X}=0\). Thus the author gives a positive answer in the class of Lindelöf spaces to A. Koyama's question: Is there a normal space \({\mathcal X}\) with dim \({\mathcal K}{\mathcal X}\neq \dim {\mathcal X}?\) The author poses the following question: Is there a Lindelöf space \({\mathcal X}\) with \(\dim{\mathcal X}=n\) and \(\dim{\mathcal K}{\mathcal X}=0\) for \(n=2,3,...,\infty\) ?