Is this a valid argument about box topology?

In summary, the conversation discusses the existence of a basis in the product space R^ω and how it relates to the box topology. The sequence x_n = {1/n, 1/, 1/n, ..., 1/n, ... } is used to show the difference between the two topologies. It is argued that there is no open set in the box topology that contains (-δ, δ) in R, hence no function converges. However, the conversation also mentions that there is no requirement for a sequence in a topological space to converge to an open set. The question of whether the basis can converge is brought up, but it is clarified that only sequences, nets, and filters can converge in topology.
  • #1
Pippi
18
0
Given the following sequence in the product space R^ω, such that the coordinates of x_n are 1/n,

x1 = {1, 1, 1, ...}
x2 = {1/2, 1/2, 1/2, ...}
x3 = {1/3, 1/3, 1/3, ...}
...

the basis in the box topology can be written as ∏(-1/n, 1/n). However, as n becomes infinitely large, the basis converges to ∏(0, 0), which is a single set and not open. Therefore the sequence is not convergent in box topology.

Since there exists a basis that converges to ∏(x, x), for any element x of R^ω, a sequence in box topology does not converge to any element in R^ω.
 
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  • #2
Pippi said:
However, as n becomes infinitely large, the basis converges to ∏(0, 0), which is a single set and not open. Therefore the sequence is not convergent in box topology.

To have a basis [itex] B [/itex] for [itex] R^\omega [/itex] don't you need to be able to represent any set in [itex] R^\omega [/itex] as a union of sets in [itex] B [/itex] , not merely the sets that are near {0,0,...}?

What kind of convergence are you talking about? Are you talking about a sequence of sets or a sequence of points? Under the usual definition of convergence, a sequence of points in a toplogical space that converges will converge to a point. There is no requirement that it converge to an open set.
 
  • #3
I don't know the right terminology. x_n represents a point in R^ω that has infinite number of coordinates. I want to use the sequence that if each of the coordinate converges as n grows large, x_n converges to a point.

I want to show the difference between product topology and box topology using the sequence x_n = {1/n, 1/, 1/n, ..., 1/n, ... }. A textbook argument, if I read correctly, says that because 1/n eventually goes to 0, there is no open set in the box topology that contains (-δ, δ) in R, hence no function converges. Am I on the right track?
 
  • #4
To get a clear answer, you're going to have state a clear question. You mention a sequence of points and then you talk about a function converging without explaining what function you mean.

If there is a textbook argument, then quote the argument. Quote it, don't just give a mangled summary. (Perhaps the discipline of copying it will make it clearer to you.)
 
  • #5
Thanks but no thank you. You are not being helpful at all.
 
  • #6
Pippi said:
Thanks but no thank you. You are not being helpful at all.

That is because your question is a bit weird. What does it mean for a basis to converge?? The only things which can converge in topology are sequences, nets and filters. Things like basises can't converge. Except if you're talking about a filter basis, but even then the OP makes little sense.
 

1. What is a box topology?

A box topology is a type of topological space in which the open sets are defined as a Cartesian product of open sets from the individual dimensions. This type of topology is commonly used in infinite-dimensional spaces.

2. How is a box topology different from other types of topologies?

Unlike other types of topologies, such as the Euclidean topology or the metric topology, the box topology does not depend on a distance metric. Instead, it is defined purely in terms of the Cartesian product of open sets.

3. What makes an argument about box topology valid?

An argument about box topology is considered valid if it follows the rules of logical reasoning and is supported by evidence and mathematical principles. It should also be clear and well-structured, with premises that lead to a logical conclusion about the box topology.

4. How can one determine if an argument about box topology is valid?

To determine if an argument about box topology is valid, one must carefully examine the premises and the conclusion. The premises should be true and relevant to the topic, and the conclusion should logically follow from the premises.

5. Are there any common misconceptions about box topology?

One common misconception about box topology is that it only applies to finite-dimensional spaces. In reality, it can also be used in infinite-dimensional spaces, making it a valuable tool in many areas of mathematics and physics.

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