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Using completions of certain universal enveloping algebras, we provide a natural setting for families of defining relations for the principal subspaces of standard modules for untwisted affine Lie algebras. We also use the theory of vertex operator algebras and intertwining operators to construct exact sequences among principal subspaces of certain standard $\widehat{\mathfrak{sl}(n)}$-modules, $n \ge 3$. As a consequence, we obtain the multigraded dimensions of the principal subspaces $W(k_1\Lambda_1 + k_2 \Lambda_2)$ and $W(k_{n-2}\Lambda_{n-2} + k_{n-1} \Lambda_{n-1})$. This generalizes earlier work by Calinescu on principal subspaces of standard $\widehat{\mathfrak{sl}(3)}$-modules.

Principal subspaces of higher-level standard $\widehat{\mathfrak{sl}(n)}$-modules