Lexicographic Square, topology

In summary, 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."
  • #1
mathsss2
38
0
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]
 
Physics news on Phys.org
  • #2
What do your base elements look like? From that it should be obvious.
 
  • #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.
 
  • #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].
 
  • #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.
 
  • #6
What base elements contain the point (a, 1)?
 
  • #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?
 
  • #8
Ahh, I completely ignored that typo, already knowing what the lexicographic order is. Everything I said holds. Can you figure it out now? :)
 
  • #9
Yes, I solved it. Thanks for the help.
 

Related to Lexicographic Square, topology

1. What is a Lexicographic Square?

A Lexicographic Square is a mathematical construct used in topology to describe certain types of spaces. It is formed by taking two copies of the real line and arranging them in a square shape, with each point on the horizontal line corresponding to a point on the vertical line.

2. How is a Lexicographic Square different from a Cartesian Product?

While a Lexicographic Square and a Cartesian Product may appear similar, they have different topological properties. In a Lexicographic Square, the open sets are determined by the horizontal and vertical lines, while in a Cartesian Product, the open sets are determined by the individual lines. This results in a different topological structure for each space.

3. What is the significance of a Lexicographic Square in topology?

The Lexicographic Square is often used as an example of a space that is connected but not path-connected. This means that there is no continuous path that connects any two points in the space. It also has other interesting properties that make it a useful tool in topology for studying different topological concepts.

4. Can a Lexicographic Square be embedded in higher dimensions?

No, a Lexicographic Square is a two-dimensional space and cannot be embedded in higher dimensions. This is because any additional dimensions would introduce new open sets and change the topological structure of the space.

5. Are there any real-world applications of a Lexicographic Square?

While the Lexicographic Square may not have direct real-world applications, it is a valuable tool in topology for understanding and studying more complex topological spaces. It can also be used to illustrate topological concepts and properties in a more visual and tangible way.

Similar threads

I Is [0,1] under the subspace topology Hausdorff, Compact, or Connected?
    • Topology and Analysis
Replies
15
Views
2K
I The Order Topology .... .... Singh, Example 1.4.4 .... ....
    • Topology and Analysis
Replies
2
Views
2K
I Chart coordinate maps of topological manifolds
    • Differential Geometry
Replies
1
Views
2K
I Discrete Topology and Closed Sets
    • Topology and Analysis
Replies
12
Views
4K
A What Topological Vector Spaces have an uncountable Schauder basis?
    • Topology and Analysis
Replies
6
Views
1K
B Solving for integral curves- how to account for changing charts?
    • Differential Geometry
Replies
4
Views
1K
B Complete the square, yes, but what does it mean?
    • General Math
Replies
19
Views
2K
The joint probability - two coins
    • Precalculus Mathematics Homework Help
Replies
6
Views
1K
Find the volume bounded by hyperboloid and plane z = ± d
    • Calculus and Beyond Homework Help
Replies
14
Views
756
I Subbasis for a Topology .... Singh, Section 1.4 ....
    • Topology and Analysis
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top