Lexicographic Square, topology

[mathsss2]

Show that any basic open set about a point on the "top edge," that is, a point of form $$(a, 1)$$, where $$a < 1$$, must intersect the "bottom edge."

Background:

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

We follow usual customs for intervals, so that $$[(a,b),(c,d)) = \{ (x,y) \in X : (a,b) \leq (x,y) < (c,d) \}$$. A subbase for the order topology on $$X$$ is the collection of all sets of form $$[(0,0),(a,b))$$ or of form $$[(a,b),(1,1)).$$

 

[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 $$[(0, 0), a)$$, $$(a, (1, 1)]$$, or $$(a, b)$$, where $$(0, 0) < a < b < (1, 1)$$.

 

[mathsss2]

So, we know the base elements are intervals of the form $$[(0, 0), a) , (a, (1, 1)]$$, or $$(a, b)$$, where $$(0, 0) < a < b < (1, 1)$$.

We need to show that any basic open set about a point on the "top edge," that is, a point of form $$(a, 1)$$, where $$a < 1$$, 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 $$(a,b)<(c,d)$$ if and only if $$a<c$$ or $$a=c$$ and $$b<d$$. 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.