This article is a fundamental study in computable analysis. In the frameworkof Type-2 effectivity, TTE, we investigate computability aspects on finite andinfinite products of effective topological spaces. For obtaining uniformresults we introduce natural multi-representations of the class of alleffective topological spaces, of their points, of their subsets and of theircompact subsets. We show that the binary, finite and countable productoperations on effective topological spaces are computable. For spaces withnon-empty base sets the factors can be retrieved from the products. We studycomputability of the product operations on points, on arbitrary subsets and oncompact subsets. For the case of compact sets the results are uniformlycomputable versions of Tychonoff's Theorem (stating that every Cartesianproduct of compact spaces is compact) for both, the cover multi-representationand the "minimal cover" multi-representation.
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