This paper answers a long-standing and important question about the existence of ultrafilters with special topological properties in the space of ultrafilters (over a discrete space). The main result is to prove that for each regular cardinal \(\kappa\), there is a uniform ultrafilter x on \(\kappa\) which is not a limit in \(\beta \kappa\) of any set of fewer than \(\kappa^+\)-many uniform ultrafilters on \(\kappa\), a so-called weak \(P_{\kappa^{++}}\)-point in the subspace \(u(\kappa)\) of uniform ultrafilters on \(\kappa\). The paper has a very informative and interesting introduction which reviews the history of the problem and the relationship to \(\kappa^+\)-good ultrafilters. The key idea of the proof has its origins in an earlier proof that a \(\kappa^{++}\)-good ultrafilter is a weak \(P_{\kappa^{++}}\)-point but it was known to be consistent that there were no \(\kappa^{++}\)-good ultrafilters on \(\kappa\). The first author’s thesis apparently includes the very interesting result that it the existence of \(\kappa^{++}\)-good ultrafilters just follows from some cardinal arithmetic. The authors isolate the key step in the above mentioned proof and introduce a new combinatorial, or basis, property of an ultrafilter, a mediocre point. Finally, a remarkable step is to greatly generalize the notion of independent family and to produce such an independent matrix. The actual construction of a mediocre point from the matrix is similar to earlier constructions (of the second author).