Results 1 to 5 of 5

Math Help - Is the product of two open sets open?

  1. #1
    Newbie
    Joined
    Sep 2008
    Posts
    12

    Is the product of two open sets open?

    Say we have open sets from two topological spaces. A is an open subset of T1 and B is an open subset of T2. So for these two open sets, A, B. Is A X B open itself? I see this is the case in R x R, where R is the real number line. I am wanting to say that this is just true in general... If so, anyone know a good proof? Or can anyway tell me how to show this?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by JohnStaphin View Post
    Say we have open sets from two topological spaces. A is an open subset of T1 and B is an open subset of T2. So for these two open sets, A, B. Is A X B open itself? I see this is the case in R x R, where R is the real number line. I am wanting to say that this is just true in general... If so, anyone know a good proof? Or can anyway tell me how to show this?
    really? it's true in \mathbb{R} \times \mathbb{R}? how so?
    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 Jhevon View Post
    really? it's true in \mathbb{R} \times \mathbb{R}? how so?
    Let (a,b) \in A\times B. Now a\in A and b\in B so there is (a-\epsilon,a+\epsilon)\subset A and (b-\epsilon,b+\epsilon)\subset B. Thus, (a-\epsilon,a+\epsilon) \times (b-\epsilon,b+\epsilon) \subset A\times B. Thus, the open square centered at (a,b) is contained in A\times B. Therefore, A\times B is open.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by ThePerfectHacker View Post
    Let (a,b) \in A\times B. Now a\in A and b\in B so there is (a-\epsilon,a+\epsilon)\subset A and (b-\epsilon,b+\epsilon)\subset B. Thus, (a-\epsilon,a+\epsilon) \times (b-\epsilon,b+\epsilon) \subset A\times B. Thus, the open square centered at (a,b) is contained in A\times B. Therefore, A\times B is open.
    ah, yes, thanks. i was thinking of open intervals as an example, but when i drew a picture to see if it were true, i treated the intervals as closed intervals
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    May 2008
    Posts
    33
    When you have two topological spaces and you want to create their product space, you need to define what is an open set.
    usually the definition is that the topological basis is the set of products of open sets, meaning that in AxB the basis contains sets of the form UxV where U is open in A and V is open in B. any other open set in AxB is a union of sets from the basis.
    so by this definition, a product of two open sets is an open set.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: September 17th 2011, 03:44 PM
  2. Metric spaces, open sets, and closed sets
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: March 16th 2011, 05:17 PM
  3. Replies: 1
    Last Post: October 30th 2010, 01:50 PM
  4. Replies: 2
    Last Post: September 26th 2010, 03:33 PM
  5. Open Sets in the Product Toplogy
    Posted in the Advanced Math Topics Forum
    Replies: 2
    Last Post: December 7th 2007, 07:52 AM

Search Tags


/mathhelpforum @mathhelpforum