Let \(Y\) be a set and \(\tau\) be a topology on a subset \(X\) of \(Y\). For each \(a\in Y\), let \(s(a)\) be a \(\tau\)-open filter on \(X\); in particular, for each \(a\in X\) let \(s(a)\) be the neighborhood filter of \(a\). Among the topologies \(\tau'\) on \(Y\) for which every \(s(a)\) is the trace of the corresponding \(\tau'\)-neighborhood filter on \(Y\), there is a coarsest topology, called the strict extension of \(\tau\), and a finest topology, called the loose extension of \(\tau\). For a given quasi-uniformity \({\mathcal U}\) on \(X\), the author calls a family of \({\mathcal U}\)-round filters \(\{s(a)\): \(a\in Y\}\) a trace filter system provided that for each \(a \in X\) \(s(a)\) is the \(\tau({\mathcal U})\)-neighborhood filter of a. A trace filter system \(S\) is admissible provided that each \(D\)-Cauchy filter in \((X,{\mathcal U})\) is finer than some member of \(S\). Using admissible trace filter systems, the author finds \(D\)-complete loose and strict extensions. For a given trace filter system on \(Y\) and quasi-uniformity \({\mathcal U}\) on \(X\), there need not exist an extension of \({\mathcal U}\) that is compatible with the strict extension of \(\tau({\mathcal U})\). The author shows, however, that if \({\mathcal U}'\) is an extension of \({\mathcal U}\) such that \(\tau({\mathcal U}')\) is the strict extension of \(\tau({\mathcal U})\) and \(X\) is dense with respect to both \(\tau({\mathcal U}')\) and \(\tau({\mathcal U}^{\prime-1})\) then \({\mathcal U}'\) is \(D\)-complete. Moreover, given any trace filter system \(S\) and quasi-uniformity \({\mathcal U}\), there is a canonical extension \({\mathcal U}'\) of \({\mathcal U}\) such that \(\tau({\mathcal U}')\) is the loose extension of \(\tau({\mathcal U})\) with respect to \(S\) and \({\mathcal U}\), and the author shows that if \(S\) is composed of \(D\)-Cauchy filters then this canonical extension, the uniformly loose extension of \({\mathcal U}\), is \(D\)-complete. Consequently, every quasi-uniform space has a \(D\)- complete uniformly loose extension. These, and the other results of the paper are well demarcated by examples, which are of interest in their own right.