Given the following sequence in the product space R^ω, such that the coordinates of x_n are 1/n,(adsbygoogle = window.adsbygoogle || []).push({});

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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# Is this a valid argument about box topology?

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