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Definition. A basis for the box topology on lime,1 X5, is the collection of all sets of the form Hag Um where UGE is open in X“ for each a. Every

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Definition. A basis for the box topology on lime,1 X5, is the collection of all sets of the
form Hag Um where UGE is open in X“ for each a. Every open set in the product topology is open in the box topology, but not Vice versa. Thus the box topology is finer than the product topology.
The next exercise and a few exercises in future chapters will show that the box topology is actually quite weird, while the product topology creates more natural spaces. Exercise 3.41. Let IR‘” be the countable product of copies of IR. So every point in R“ is a
sequence (x1, x2, x3, …). LetA C [Rm be the set consisting of all points with only positive coordinates. Show that in the product topology, 0 = (0, 0,0,…) is a limit point of the
set A, and there is a sequence of points in A converging to 0. Then show that in the box topology, 0 = (0, 0, 0,…) is a limit point of the set A, but there is no sequence of points in
A converging to 0. Exercise 3.42. Show that the set 2M with the box topology is a discrete space, whereas the
set 2Nl with the product topology has no isolated points.

 
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