1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Lexicographic Square, topology

  1. Dec 2, 2008 #1
    Show that any basic open set about a point on the "top edge," that is, a point of form [tex](a, 1)[/tex], where [tex]a < 1[/tex], must intersect the "bottom edge."

    Background:

    Definition- The lexicographic square is the set [tex]X = [0,1] \times [0,1][/tex] with the dictionary, or lexicographic, order. That is [tex](a, b) < (c, d)[/tex] if and only if either [tex]a < b[/tex], or [tex]a = b[/tex] and [tex]c < d[/tex]. This is a linear order on [tex]X[/tex], and the example we seek is [tex]X[/tex] with the order topology.

    We follow usual customs for intervals, so that [tex][(a,b),(c,d)) = \{ (x,y) \in X : (a,b) \leq (x,y) < (c,d) \}[/tex]. A subbase for the order topology on [tex]X[/tex] is the collection of all sets of form [tex][(0,0),(a,b))[/tex] or of form [tex][(a,b),(1,1)).[/tex]
     
  2. jcsd
  3. Dec 2, 2008 #2
    What do your base elements look like? From that it should be obvious.
     
  4. Dec 4, 2008 #3
    This problem is sort of confusing me. I am not sure what the base elements look like here. What do they look like? Maybe I am just not seeing something.
     
  5. Dec 4, 2008 #4
    The base elements are all finite intersections of your subbase elements; they are intervals of the form [tex][(0, 0), a)[/tex], [tex](a, (1, 1)][/tex], or [tex](a, b)[/tex], where [tex](0, 0) < a < b < (1, 1)[/tex].
     
  6. Dec 7, 2008 #5
    So, we know the base elements are intervals of the form [tex][(0, 0), a) , (a, (1, 1)][/tex], or [tex](a, b)[/tex], where [tex](0, 0) < a < b < (1, 1)[/tex].

    We need to show that any basic open set about a point on the "top edge," that is, a point of form [tex](a, 1)[/tex], where [tex]a < 1[/tex], must intersect the "bottom edge."

    How is this obvious now? I don't understand the connection? Thanks for all your help with topology, I was able to solve the other problem you helped me with too.
     
  7. Dec 7, 2008 #6
    What base elements contain the point (a, 1)?
     
  8. Dec 8, 2008 #7
    Turns out there was a typo in the problem [that was throwing me off a lot]. So, the lexicographic order should be [tex](a,b)<(c,d)[/tex] if and only if [tex]a<c[/tex] or [tex]a=c[/tex] and [tex]b<d[/tex]. So, is our solution the same knowing this now?
     
  9. Dec 8, 2008 #8
    Ahh, I completely ignored that typo, already knowing what the lexicographic order is. Everything I said holds. Can you figure it out now? :)
     
  10. Dec 8, 2008 #9
    Yes, I solved it. Thanks for the help.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Lexicographic Square, topology
  1. DNA Topology (Replies: 2)

  2. Problem in topology (Replies: 15)

  3. Differential Topology (Replies: 0)

Loading...