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Lexicographic Square, topology 
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#1
Dec208, 08:57 PM

P: 38

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
Dec208, 09:46 PM

P: 534

What do your base elements look like? From that it should be obvious.



#3
Dec408, 12:05 AM

P: 38

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.



#4
Dec408, 12:23 AM

P: 534

Lexicographic Square, topology
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].



#5
Dec708, 02:36 PM

P: 38

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. 


#6
Dec708, 02:41 PM

P: 534

What base elements contain the point (a, 1)?



#7
Dec808, 02:54 PM

P: 38

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?



#8
Dec808, 03:48 PM

P: 534

Ahh, I completely ignored that typo, already knowing what the lexicographic order is. Everything I said holds. Can you figure it out now? :)



#9
Dec808, 07:42 PM

P: 38

Yes, I solved it. Thanks for the help.



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