The quasi-metric of complexity convergence

For any weightable quasi-metric space (X, d) having a maximum with respect to the associated order ≤_d, the notion of the quasi-metric of complexity convergence on the the function space (equivalently, the space of sequences) X^ω, is introduced and studied. We observe that its induced quasi-uniformity is finer than the quasi-uniformity of pointwise convergence and weaker than the quasi-uniformity of uniform convergence. We show that it coincides with the quasi-uniformity of pointwise convergence if and only if the quasi-metric space (X, d) is bounded and it coincides with the quasi-uniformity of uniform convergence if and only if X is a singleton. We also investigate completeness of the quasi-metric of complexity convergence. Finally, we obtain versions of the celebrated Grothendieck theorem in this context.