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Math Help - (Very) Easy Topology

  1. #1
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    (Very) Easy Topology

    Can someone please describe the standard topology on the closed interval [0,1]? It seems (to me) like it should consist of the open balls, {0}, and {1} (and the union/finite intersection of these, etc.)... Thanks.
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    Quote Originally Posted by sleepingcat View Post
    Can someone please describe the standard topology on the closed interval [0,1]?
    If there is a standard topology on [0,1] it has countable subbasis.
    So what do you mean by standard?
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    Quote Originally Posted by sleepingcat View Post
    Can someone please describe the standard topology on the closed interval [0,1]? It seems (to me) like it should consist of the open balls, {0}, and {1} (and the union/finite intersection of these, etc.)... Thanks.
    I assume you mean a subset topology.

    Since [0,1] is a subset of \mathbb{R} it means U\subseteq [0,1] is open iff U = [0,1] \cap V for some open subset V of \mathbb{R}.

    Therefore, the open sets of [0,1] under the subspace topology are (a,b), [0,a), (a,1] where 0<a<b<1.
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    Hello,
    Quote Originally Posted by sleepingcat View Post
    Can someone please describe the standard topology on the closed interval [0,1]? It seems (to me) like it should consist of the open balls, {0}, and {1} (and the union/finite intersection of these, etc.)... Thanks.
    You don't talk about balls in this dimension oO

    The standard topology of [0,1] may be the subspace topology on [0,1] \subset \mathbb{R}

    The standard topology on [0,1] may then be \tau=\{\mathcal{O} \cap [0,1] ~ : ~ \mathcal{O} \in \tau_{\mathbb{R}} \}

    And \tau_{\mathbb{R}}=\{]a,b[ ~ : ~ a<b, ~a \in \mathbb{R} \cup \{-\infty\} \text{ and } b \in \mathbb{R} \cup \{+\infty\}\}


    So \tau=\{[0,a[ ~,~ ]b,1] ~,~ ]a,b[ ~:~ 0<a \le 1 \text{ and } 0 \le b < 1 \}
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