6533b828fe1ef96bd1289105

RESEARCH PRODUCT

Some Remarks about Product Spaces

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Summary This article covers some technical aspects about the product topology which are usually not given much of a thought in mathematics and standard literature like [7] and [6], not even by Bourbaki in [4]. Let {Ti}i∈I be a family of topological spaces. The prebasis of the product space T = ∏ i∈I Ti is defined in [5] as the set of all π −1 i (V) with i ∈ I and V open in Ti . Here it is shown that the basis generated by this prebasis consists exactly of the sets ∏ i∈I Vi with Vi open in Ti and for all but finitely many i ∈ I holds Vi = Ti . Given I = {a} we have T ≅ Ta , given I = {a, b} with a≠ b we have T ≅ Ta ×Tb . Given another family of topological spaces {Si}i∈I such that Si ≅ Ti for all i ∈ I, we have S = ∏ i∈I Si ≅ T. If instead Si is a subspace of Ti for each i ∈ I, then S is a subspace of T. These results are obvious for mathematicians, but formally proven here by means of the Mizar system [3], [2].