Is this a valid argument about box topology?

[Pippi]

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.

 

[Stephen Tashi]

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 $B$ for $R^\omega$ don't you need to be able to represent any set in $R^\omega$ as a union of sets in $B$ , 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.

 

[Pippi]

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?

 

[Stephen Tashi]

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.)

 

[Pippi]

Thanks but no thank you. You are not being helpful at all.

 

[micromass]

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.