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I'm looking for some help in understanding one of the theorems stated in section 20 of Munkres. The theorem is as follows:

The uniform topology on ##\mathbb{R}^J## (where ##J## is some arbitrary index set) is finer than the product topology and coarser than the box topology; these three topologies are all different if ##J## is infinite.

This theorem seems to break down into two cases: ##J## finite, or ##J## infinite. In the case that ##J## is finite, aren't the box and product topologies equivalent? Hence it seems like the first sentence of the theorem can be strengthened to say all three topologies are equivalent?

For the second case, if ##J## is infinite, I thought that the box topology is finer than the product topology, since the product topology has the restriction that for each basis element, only finitely many of each of the ##\mathbb{R}_i## are open such that they are not equal to ##\mathbb{R}_i## itself, whereas the box topology does not have this restriction. So I'm not sure I understand why these topologies are different in the case that ##J## is infinite? All the statements in this theorem seem to contradict what I understood from the previous chapter on product topologies.

Any help explaining the theorem is appreciated. Thanks!

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# Explanation of uniform topology theorem in Munkres

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