- #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.
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.