This article is a fundamental study in computable analysis. In the framework of Type-2 effectivity, TTE, we investigate computability aspects on finite and infinite products of effective topological spaces. For obtaining uniform results we introduce natural multi-representations of the class of all effective topological spaces, of their points, of their subsets and of their compact subsets. We show that the binary, finite and countable product operations on effective topological spaces are computable. For spaces with non-empty base sets the factors can be retrieved from the products. We study computability of the product operations on points, on arbitrary subsets and on compact subsets. For the case of compact sets the results are uniformly computable versions of Tychonoff's Theorem (stating that every Cartesian product of compact spaces is compact) for both, the cover multi-representation and the "minimal cover" multi-representation.