Explanation of uniform topology theorem in Munkres

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Discussion Overview

The discussion revolves around the uniform topology theorem as presented in section 20 of Munkres' text. Participants are exploring the relationships between the uniform topology, product topology, and box topology on the space ##\mathbb{R}^J##, considering both finite and infinite cases of the index set ##J##. The conversation includes clarifications on definitions and the implications of the theorem.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions whether the box and product topologies are equivalent when ##J## is finite, suggesting that the theorem could be strengthened to state all three topologies are equivalent in that case.
  • Another participant agrees with the assertion about the equivalence of the box and product topologies when ##J## is finite.
  • In the case of infinite ##J##, a participant expresses confusion about the relationship between the box topology and product topology, asserting that the box topology is finer due to the lack of restrictions present in the product topology.
  • There is a clarification that the box topology is indeed finer than the uniform topology, which in turn is finer than the product topology.
  • A participant reflects on the definition of "different" in the context of topologies, indicating a possible misunderstanding of the terminology used in the theorem.
  • Another participant inquires about the nature of the uniform topology, questioning if it refers to the topology of uniform convergence in function spaces.

Areas of Agreement / Disagreement

Participants generally agree on the equivalence of the box and product topologies when ##J## is finite. However, there remains uncertainty and debate regarding the distinctions between the topologies when ##J## is infinite, as well as the interpretation of the term "different" in the theorem.

Contextual Notes

Participants express confusion regarding the definitions and implications of the topologies involved, particularly in relation to the restrictions of the product topology compared to the box topology. There is also a lack of clarity on the terminology used in the theorem, which may affect understanding.

mathmonkey
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Hi all,

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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mathmonkey said:
Hi all,

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?

Correct.

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.

Correct.

So what we're claiming is that the box topology is finer than the uniform topology, and that the uniform topology is finer than the product topology.
 
Oh...I could've sworn I read somewhere earlier in the text that two topologies are defined to be different if neither is finer or coarser than the other. So I suppose "different" in this case just means they are not equal?

I guess I just bashed my head over the table for an hour over misunderstanding of wording then :redface: . At least its good to see my previous understanding of the topic wasn't wrong though.
 
Just curious, I don't have Munkres --nor his book :) --with me. Is the uniform topology

the topology of uniform convergence in function spaces?
 

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