Results 1 to 3 of 3

Math Help - Trouble with a product topology problem

  1. #1
    Junior Member
    Joined
    Feb 2010
    Posts
    37

    Trouble with a product topology problem

    Hello, This is my problem: let d:X\times X\rightarrow R be a distance on X. We know d is continous with the prodcut topology when the topology given to X is the metric topology. Show tha if d is continous with the product topology when the topology given to X is \mathcal{F} then this topology is finner than the metric toplogy.
    Attempts: we know that there is a coarsest topology for which d is continous is \{d^{-1}(A):A open set in R \} I tried to prove that this topology is the product topology of the metric but this doesn't seem to be the case.
    I also tried to prove directly that if U is open in the metric topology then U is in \mathcal{F}
    Any comments will be appreciated.
    Last edited by facenian; August 24th 2010 at 03:50 AM. Reason: gramma correction
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Aug 2010
    Posts
    44
    The open balls B(x_{0},\epsilon) := \{ y \in X : d(x_{0},y)<\epsilon \} are a basis for the topology on X, which is induced by the metric d.
    It is therefor sufficient to show that every open ball is also in \mathcal{F}.

    For every x_{0} \in X consider the map

    h: X \rightarrow \mathbb{R}
    h(x):=d(x,x_0)
    , which is the composition of
    f:X \rightarrow X \times X
    , f(x):=(x,x_{0})
    and
    d:X\times X \rightarrow \mathbb{R}<br />

    both are continious in the topology \mathcal{F}, so the composition is also continous and we have
    B(x_{0},\epsilon)=h^{-1}((-\infty , \epsilon)) \in \mathcal{F}
    Last edited by Iondor; August 25th 2010 at 02:57 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Feb 2010
    Posts
    37
    I think you solved my problem, thanks
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] basic topology trouble
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: August 5th 2011, 11:53 AM
  2. trouble with inner product space problem...
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: November 18th 2009, 07:27 PM
  3. discrete topology, product topology
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: December 13th 2008, 02:19 PM
  4. discrete topology, product topology
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: December 7th 2008, 01:01 PM
  5. trouble understanding the relative topology and subspaces
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: June 5th 2008, 09:38 PM

Search Tags


/mathhelpforum @mathhelpforum