Lexicographic Square, topology

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Discussion Overview

The discussion revolves around the properties of the lexicographic square in topology, specifically focusing on basic open sets and their intersections with edges of the square. Participants explore the definitions and implications of the order topology on the set X = [0,1] × [0,1].

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant asks for clarification on the appearance of base elements in the context of the lexicographic square.
  • Another participant describes the base elements as finite intersections of subbase elements, specifically intervals of the form [(0, 0), a), (a, (1, 1)], or (a, b), where (0, 0) < a < b < (1, 1).
  • A participant expresses confusion about the connection between basic open sets and their intersection with the bottom edge of the square.
  • There is a correction regarding a typo in the definition of the lexicographic order, which prompts a participant to question if the solution remains valid with the corrected definition.
  • One participant asserts that their previous statements still hold despite the typo, suggesting confidence in their understanding of the lexicographic order.
  • A later reply indicates that one participant successfully solved the problem after receiving assistance.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the base elements and their implications. There is no clear consensus on the initial confusion, but some participants agree on the nature of the base elements once clarified.

Contextual Notes

The discussion includes a correction of a typo that affects the understanding of the lexicographic order, which may influence the interpretation of the problem. However, the implications of this correction remain unresolved.

Who May Find This Useful

Readers interested in topology, particularly those studying order topologies and their properties, may find this discussion relevant.

mathsss2
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Show that any basic open set about a point on the "top edge," that is, a point of form (a, 1), where a &lt; 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) &lt; (c, d) if and only if either a &lt; b, or a = b and c &lt; 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) &lt; (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)).
 
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What do your base elements look like? From that it should be obvious.
 
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.
 
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) &lt; a &lt; b &lt; (1, 1).
 
So, we know the base elements are intervals of the form [(0, 0), a) , (a, (1, 1)], or (a, b), where (0, 0) &lt; a &lt; b &lt; (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 &lt; 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.
 
What base elements contain the point (a, 1)?
 
Turns out there was a typo in the problem [that was throwing me off a lot]. So, the lexicographic order should be (a,b)&lt;(c,d) if and only if a&lt;c or a=c and b&lt;d. So, is our solution the same knowing this now?
 
Ahh, I completely ignored that typo, already knowing what the lexicographic order is. Everything I said holds. Can you figure it out now? :)
 
Yes, I solved it. Thanks for the help.
 

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