Abstract
We prove that the set of homotopy classes of the paths in a topological ring is a topological ringobject (called topological ring-groupoid). Let p : X → X be a covering map and let X be a topological ring.We define a category UTRCov(X) of coverings of X in which both X and X have universal coverings, and acategory UTRGdCov( π1X ) of coverings of topological ring-groupoid π1X , in which X and R0 = X haveuniversal coverings, and then prove the equivalence of these categories. We also prove that the topologicalring structure of a topological ring-groupoid lifts to a universal topological covering groupoid.
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