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The open sets in the product topology are unions (finite or infinite) of sets of the form <math>\prod_{i\in I} U_i</math>, where each ''U<sub>i</sub>'' is open in ''X<sub>i</sub>'' and ''U''<sub>''i''</sub> ≠ ''X''<sub>''i''</sub> for only finitely many ''i''. In particular, for a finite product (in particular, for the product of two topological spaces), the products of base elements of the ''X<sub>i</sub>'' gives a basis for the product <math>\prod_{i\in I} X_i</math>.
The product topology on ''X'' is the topology generated by sets of the form ''p<sub>i</sub>''<sup>−1</sup>(''U<sub>i</sub>''), where ''i'' is in ''I '' and ''U<sub>i</sub>'' is an open subset of ''X<sub>i</sub>''. In other words, the sets {''p<sub>i</sub>''<sup>−1</sup>(''U<sub>i</sub>'')} form a [[subbase]] for the topology on ''X''. A [[subset]] of ''X'' is open if and only if it is a (possibly infinite) [[union (set theory)|union]] of [[intersection (set theory)|intersections]] of finitely many sets of the form ''p<sub>i</sub>''<sup>−1</sup>(''U<sub>i</sub>''). The ''p<sub>i</sub>''<sup>−1</sup>(''U<sub>i</sub>'') are sometimes called [[open cylinder]]s, and their intersections are [[cylinder set]]s.
In general, the product of the topologies of each ''X<sub>i</sub>'' forms a basis for what is called the [[box topology]] on ''X''. In general, the box topology is [[finer topology|finer]] than the product topology, but for finite products they coincide.
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