Uniform Metric vs Box Topology

In summary, the conversation discusses the uniform metric on \mathbb{R}^\omega and defines U(x,\epsilon) as a product subset of \mathbb{R}^\omega. The conversation then goes on to discuss two points, x and y, in \mathbb{R}^\omega and the distance between them in the uniform metric, denoted by \rho(x,y). The conversation raises questions about the properties of the ball of radius \epsilon about x, denoted by B_\rho(x,\epsilon), and U(x,\epsilon). It is shown that there exists a y\in U(x,\epsilon) such that \rho(x,y) = \epsilon, making it clear that y\not\in B_\
  • #1
joeboo
35
0
Let [itex]\rho[/itex] be the uniform metric on [itex]\mathbb{R}^\omega[/itex]
For reference, for two points:
[tex]x = (x_i)[/tex] and [tex]y = (y_i)[/tex] in [tex]\mathbb{R}^\omega[/tex]

[tex]\rho(x,y) = \sup_i\{ \min\{|x_i - y_i|, 1\}\}[/tex]

Now, define:

[tex]U(x,\epsilon) = \prod_i{(x_i - \epsilon, x_i + \epsilon)} \subset \mathbb{R}^\omega[/tex]

I need to show 2 things:
1) For [itex]0 < \epsilon < 1[/itex], the ball of radius [itex]\epsilon[/itex] about [itex]x[/itex] in the uniform metric, [itex]B_\rho(x,\epsilon) \neq U(x,\epsilon)[/itex]

2)[tex]B_\rho(x,\epsilon) = \bigcup_{\delta < \epsilon}{U(x,\delta)}[/tex]

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Now, first off, I can't understand how 1 can be true. It makes no sense to me. Am I misunderstanding the metric? To further obfuscate the issue, 2 seems to contradict 1. Afterall, doesn't:

[tex]\bigcup_{\delta < \epsilon}{\prod_i{(x_i-\delta, x_i+\delta)}}} = \prod_i{\bigcup_{\delta<\epsilon}{(x_i-\delta,x_i+\delta)}}}[/tex]

The union inside the product of the latter being [itex](x_i-\epsilon, x_i+\epsilon)[/itex]? Thus, isn't part 2 suggesting that, in fact, [itex]B_\rho(x,\epsilon) = U(x,\epsilon)[/itex]

Am I missing something big here? I feel like it's obvious the problem is incorrect, so I can't understand how the author ( Munkres' Topology First Edition, section 2-9 #6 for reference ) could make the mistake. Please, someone throw me a bone here, I'm just stumped
 
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  • #2
This is actually a good problem because it exposes some of the subtlety that can be in the supremum. I claim that there is a [tex]y\in U(x,\epsilon)[/tex] such that [tex]\rho(x,y) =\epsilon[/tex] so [tex]y\not\in B_\rho (x,\epsilon)[/tex]. Namely [tex]y_i =x_i +\frac{i}{i+1}\epsilon [/tex]. You should be able to show that such a [tex]y[/tex] satisfies what I said above. Of course I'm assuming you have a countably infinite index set but you can work around that if you don't want that stipulation.

Hopefully with that idea of the supremum in mind you should be able to see why
[tex]\bigcup_{\delta < \epsilon}{\prod_i{(x_i-\delta, x_i+\delta)}}} \ne \prod_i{\bigcup_{\delta<\epsilon}{(x_i-\delta,x_i+\delta)}}}[/tex]

Hope that's some help,
Steven
 
  • #3
Haha!

Perfect, snoble, thank you.
Now that I see it, I can't believe what I was thinking.
I think at some point, I was equating the open ball with the closed ball in the uniform tolology. As in, it included the values for which the supremum was reached. Obviously, that's not the definition of a basis element for the uniform metric topology. Silly me :redface:

In any event, it's now painfully clear to me that the uniform topology is an entirely distinct animal from the product and box topologies. Now to try and wrap my head around it.

Thanks again for pointing out my broken line of thought.
 

Related to Uniform Metric vs Box Topology

1. What is the difference between Uniform Metric and Box Topology?

The main difference between Uniform Metric and Box Topology is that Uniform Metric is a metric space, while Box Topology is a topological space. In Uniform Metric, the distance between any two points is defined by a metric, while in Box Topology, the open sets are defined by the collection of open intervals in each coordinate.

2. What is the purpose of using Uniform Metric in comparison to Box Topology?

The purpose of using Uniform Metric is to measure the distance between points in a metric space in a consistent and uniform manner. This allows for the definition of important concepts such as continuity, convergence, and completeness. On the other hand, Box Topology is used to define a topological space, where open sets are used to determine the properties of the space.

3. Can Uniform Metric be used in place of Box Topology?

No, Uniform Metric and Box Topology serve different purposes and cannot be used interchangeably. Uniform Metric is used to define a metric space, while Box Topology is used to define a topological space. While Uniform Metric can be used to induce a topology, it may not always be equivalent to the topology defined by Box Topology.

4. What are the advantages of using Uniform Metric over Box Topology?

One advantage of using Uniform Metric is that it allows for the definition of important concepts such as continuity, convergence, and completeness. These concepts are necessary in many areas of mathematics and science. Additionally, Uniform Metric is a more rigorous and precise way of measuring distances between points compared to the open sets defined in Box Topology.

5. Can Uniform Metric and Box Topology be used together?

Yes, Uniform Metric and Box Topology can be used together in some cases. For example, Uniform Metric can be used to define a metric space, and then a topology can be induced from this metric space using the open sets defined in Box Topology. This allows for the use of both concepts in a mathematical or scientific context.

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