Topological Lifting of Mundici Equivalence and Di Nola--Lettieri Equivalence

We prove that summabilitycharacterises precisely when a topological MV-algebra can be realized asthe unit interval of a lattice-ordered topological abelian group, therebyproviding a topological lifting of both Mundici equivalence and Di Nola--Lettieri equivalence. Applicationsinclude a Birkhoff--Kakutani-type metrisation theorem, asecond-countability criterion, and a topological extension of a keycontinuity lemma from the proof theory of first-order {\L}ukasiewiczlogic---following Baaz--Metcalfe's hypersequent approach to approximatecompletenesss---and provides an $A$-valued topological semantics for the approximate strong completeness of~$\mathbf{G}\text{{\L}}\forall$, where $A$ ranges over summable topological MV-algebras.