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Math Help - Box topology

  1. #1
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    Box topology

    Hello math experts,
    Let X=R^{N} be a topological space with the standard box topology.
    Show that the collection of sequences that converge to 0 is an open-closed set in X.

    thanks
    Last edited by aharonidan; March 29th 2011 at 02:28 AM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by aharonidan View Post
    Hello math experts,
    Let X=R^n be a topological space with the standard box topology.
    Show that the collection of sequences that converge to 0 is an open-closed set in X.

    thanks
    I'm confused. Presumably you mean that X=\mathbb{R}^\mathbb{N}. I'll help you with one half. Suppose that (a_n)\notin Z where Z is the set of all null sequences. Then, there exists some \varepsilon>0 such that for every N\in\mathbb{N} one has that there is some n\geqslant N for which |a_n|>\varepsilon. So, let \displaystyle O=\prod_{n\in\mathbb{N}}A_n where A_n=\mathbb{R}-(-\varepsilon,\varepsilon) if |a_n|>\varepsilon and \mathbb{R} otherwise. This is clearly open in the box topology since it's the product of open sets, but clearly \displaystyle O\cap Z=\varnothing. Thus X-Z is open etc. etc.
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