Results 1 to 2 of 2

Math Help - Topology Question

  1. #1
    Member
    Joined
    Jan 2008
    Posts
    78

    Topology Question

    On the plane \Re^2

    \beta = {(a, b) * (c, d) \subset \Re^2  | a < b, c<d }

    (a) Show that \beta is a basis for a topology on \Re^2
    Last edited by flaming; January 18th 2009 at 05:57 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    Quote Originally Posted by flaming View Post
    On the plane \Re^2

    \beta = \{(a, b) \times (c, d) \subset \Re^2 | a < b, c<d \}

    (a) Show that \beta is a basis for a topology on \Re^2
    My attempt to this problem is as follows:

    To show \beta is a basis, we need to show that

    (1) For each x \in \Re^2, there is at least one basis element \beta containing x.

    Let x be (i,j) \in \Re^{2}, i,j \in \Re. Then there is a basis element containing x such that  (i - \frac{1}{n}, i + \frac{1}{n}) \times (j - \frac{1}{n}, j+\frac{1}{n}), n is a positive integer.

    (2) If x belongs to the intersection of two basis elements \beta_{1} and \beta_{2} and then there is a basis element \beta_{3} containg x such that \beta_{3} \subset \beta_{1} \cap \beta_{2}.

    There are several cases of intersections, we find the basis element satisfying the above.
    Let
    \beta_{1} = (a, b) \times (c, d) \subset \Re^2 , (a < b, c<d )
    \beta_{2} = (i, j) \times (p, q) \subset \Re^2 (i < j, p<q)

    For instance, a \leq i \leq b \leq j and c \leq p \leq d \leq q, and if x belongs to the above \beta_{1} \cap \beta_{2}, then x belongs to the below \beta_{3}
    \beta_{3} = (i, b) \times (p, d) \subset \Re^2 , (i < b, p<d )

    In each cases,  \beta_{3} can be described as
    \beta_{3} = (l, m) \times (n, o) \subset \Re^2 , (l < m, n<o ) .

    Thus, \beta is a basis for \Re^{2}.
    Last edited by aliceinwonderland; January 19th 2009 at 12:11 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Topology question
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: August 3rd 2011, 05:13 AM
  2. Topology question
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: March 1st 2010, 03:31 PM
  3. Topology Question...
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: November 25th 2009, 04:29 AM
  4. Topology question
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 6th 2009, 10:00 PM
  5. Topology Question
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 7th 2009, 08:25 PM

Search Tags


/mathhelpforum @mathhelpforum