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General and Differential Topology

Find conditions under which the box topology is strictly finer than the product topology. See attached file for full problem description.

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Question:

Find conditions under which the box topology is strictly finer than the product topology.

Definition of Box Topology:

Let I be an arbitrary index set and suppose X_i is a topological space for every i in I. Set X = Π X_i, the cartesian product of the sets X_i.
The box topology is generated by the collection B = { Π U_i | U_i open in X_i}, the cartesian products of every open set of every X_i.

Definition ...

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