The quasi-metric of complexity convergence
نویسندگان
چکیده
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, 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. AMS (1991) Subject classification: 54E15, 54E35, 54C35.
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