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box topology∗ yark† 2013-03-21 14:22:22 Let {(Xα , Tα )}α∈A be a family of topological spaces. Let Y denote the generalized Cartesian product of the sets Xα , that is Y Y = Xα . α∈A Let B denote the set of all products of open sets of the corresponding spaces, that is ( ) Y B= Uα Uα ∈ Tα for all α ∈ A . α∈A Now we can construct the box product (Y, S), where S, referred to as the box topology, is the topology generated by the base B. When A is a finite set, the box topology coincides with the product topology. Example As an example, the box product of two topological spaces (X0 , T0 ) and (X1 , T1 ) is (X0 × X1 , S), where the box topology S S (which is the same as the product topology) consists of all sets of the form i∈I (Ui × Vi ), where I is some index set and for each i ∈ I we have Ui ∈ T0 and Vi ∈ T1 . ∗ hBoxTopologyi created: h2013-03-21i by: hyarki version: h33095i Privacy setting: h1i hDefinitioni h54A99i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1