Results 1 to 4 of 4

Math Help - To show metrizable

  1. #1
    Junior Member
    Joined
    Aug 2009
    Posts
    62

    To show metrizable

    Show that product topology on Rd*R is metrizable..

    (Rd is discrete topology on R.)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    Quote Originally Posted by math.dj View Post
    Show that product topology on Rd*R is metrizable..

    (Rd is discrete topology on R.)
    A finite products of metrizable spaces is metrizable. A discrete topology on R is metrizable since it is induced by a discrete metric on R. A standard topology on R is also metrizable since it is induced by a standard metric on R. You can also use some metrization theorems here. You can use a Nagata-Smirnov metrization theorem to show that a discrete topology on R is metrizable and use a Urysohn metrization theorem to show that a standard topology on R is metrizable.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Aug 2009
    Posts
    62
    We have yet not come across Nagata-Smirnov metrization theorem and Urysohn metrization theorem ..so i can't use them..can u please tell me how do i show that R with usual topology is metrizable..i simply don't know how to write it..


    thank you for your help
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    Quote Originally Posted by math.dj View Post
    We have yet not come across Nagata-Smirnov metrization theorem and Urysohn metrization theorem ..so i can't use them..can u please tell me how do i show that R with usual topology is metrizable..i simply don't know how to write it..


    thank you for your help
    OK. It is straightfoward to see that a discrete metric on R induces a discrete topology on R. So you don't actually need Nagata-Smirnov metrization theorem.

    Anyhow, to show that usual topology on R is metrizable, I recommend you to use Urysohn metrization theorem. It says that "Every second-countable regular Hausdorff space is metrizable." You need to verify that a standard(usual) topology on R is second countable and regular Hausdorff.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Weakest Metrizable Topology?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: October 17th 2011, 06:56 AM
  2. Show
    Posted in the Algebra Forum
    Replies: 1
    Last Post: September 17th 2011, 09:19 PM
  3. irrationals, metrizable, Michael Line
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 20th 2009, 11:34 AM
  4. how to show show this proof using MAX
    Posted in the Calculus Forum
    Replies: 2
    Last Post: January 14th 2009, 01:05 PM
  5. show...
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 26th 2008, 11:56 AM

Search Tags


/mathhelpforum @mathhelpforum