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

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.

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

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.