Is [a,b] ever an open set in the order topology?

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Discussion Overview

The discussion revolves around the nature of the interval [a,b] in the context of the order topology, specifically questioning under what conditions this interval can be considered an open set. The scope includes theoretical exploration and clarification of concepts related to topology and order structures.

Discussion Character

  • Exploratory, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant notes that their textbook suggests [a,b] can sometimes be an open set, prompting questions about the conditions under which this is true.
  • Another participant proposes that considering the natural numbers might provide insight, suggesting that intervals like [3,5] can be viewed as open sets in this context.
  • A further contribution clarifies that in the order topology, an open set is defined as a union of open intervals, raising questions about the validity of treating [3,5] as open.
  • There is confusion expressed regarding the definition of open sets in the order topology and how it applies to specific examples.

Areas of Agreement / Disagreement

Participants express uncertainty about the classification of [a,b] as an open set, with some suggesting it can be open under certain conditions while others challenge this view, indicating a lack of consensus.

Contextual Notes

The discussion highlights the need for clarity on definitions and the conditions under which intervals are considered open in the order topology, but does not resolve these issues.

Who May Find This Useful

This discussion may be useful for students and individuals interested in topology, particularly those exploring the order topology and the properties of open sets within different mathematical structures.

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My textbook is indicating to me that sometimes {x \in X : a <= x <= b} is an open set. How can this happen?

My only guess is that if X has a smallest and largest element, called a and b, then sure. Otherwise?
 
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Let's look at the natural numbers for inspiration

[3,5]=(2,6)

[4,9]=(3,10)

etc. Hopefully the generalization becomes clear
 
So that would result from our space being the natural numbers and using the order topology. But why is that interval open? I suppose that for any element x in [3,5], we can find an open set, U, around x such that U \subset X.

For 3 we would choose the open set (2,4)?
 
But why is that interval open?

I'm confused. The definition of an open set in the order topology is that it's a union of open intervals. What do you mean here?
 

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