Lexicographic Square, topology

mathsss2

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]

adriank

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

mathsss2

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.

adriank

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].

mathsss2

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.

adriank

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

mathsss2

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?

adriank

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

mathsss2

Yes, I solved it. Thanks for the help.