Covering Multiplicity

Let \((X,\tau)\) be a nonempty topological space and \(\mathcal{U}=(U_i)_{i\in I}\) a cover of \(X\). The covering multiplicity \(\kappa\) regarding \(\mathcal{U}\) is defined by

\begin{equation*} \kappa = \sup_{y\in X}\; \#\{i\in I\mid y\in U_i\}\in \mathbb{N}\cup \{\infty\}. \end{equation*}