Results 1 to 4 of 4

Math Help - (Very) Easy Topology

  1. #1
    Newbie
    Joined
    May 2008
    Posts
    16

    (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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,617
    Thanks
    1581
    Awards
    1
    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?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    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.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    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 \}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Order topology = discrete topology on a set?
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: August 6th 2011, 11:19 AM
  2. a topology such that closed sets form a topology
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: June 14th 2011, 04:43 AM
  3. Show quotient topology on [0,1] = usual topology on [0,1]
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: November 5th 2010, 04:44 PM
  4. Easy Metric Space Topology
    Posted in the Math Challenge Problems Forum
    Replies: 4
    Last Post: May 19th 2010, 12:52 PM
  5. discrete topology, product topology
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: December 13th 2008, 02:19 PM

Search Tags


/mathhelpforum @mathhelpforum