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

In summary, the conversation discusses the concept of open sets and how they can exist in certain cases, such as when the space has a smallest and largest element. The speaker also gives examples using the natural numbers and the order topology. The conversation ends with a question about the definition of an open set.
  • #1
Zoomingout
2
0
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?
 
Physics news on Phys.org
  • #2
Let's look at the natural numbers for inspiration

[3,5]=(2,6)

[4,9]=(3,10)

etc. Hopefully the generalization becomes clear
 
  • #3
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)?
 
  • #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?
 
  • #5


Yes, it is possible for the set [a,b] to be open in the order topology. This can occur when the set X has a smallest and largest element, which we can call a and b. In this case, the set [a,b] would consist of all elements in X that are greater than or equal to a and less than or equal to b. Since there is no element in X that is both greater than b and less than a, the set [a,b] would be open in the order topology.

However, if X does not have a smallest and largest element, then [a,b] would not be open in the order topology. In this case, there would be elements in X that are greater than b or less than a, making [a,b] a closed set in the order topology.

It is important to note that in the order topology, a set can be open, closed, or neither depending on the specific set and the elements in X. Therefore, it is possible for [a,b] to be open in some cases and closed in others. This is why it is important to carefully consider the elements and properties of X when determining the openness of a set in the order topology.
 

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

What is an open set in the order topology?

In the order topology, an open set is a set that contains all the elements between any two points in the set. This means that for any element in the set, there exists a neighborhood of points in the set that surrounds it.

What is the difference between an open set and a closed set in the order topology?

An open set in the order topology contains all the points between any two points in the set, while a closed set contains its boundary points. In other words, a closed set includes its endpoints, whereas an open set does not.

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

No, [a,b] is never an open set in the order topology. This is because it contains its boundary points, a and b, and therefore does not meet the criteria of an open set in the order topology.

Can an open set in the order topology be an infinite set?

Yes, an open set in the order topology can be an infinite set. As long as the set contains all the elements between any two points, it can be considered an open set in the order topology.

How is the order topology different from other topologies?

The order topology is different from other topologies because it is based on the ordering of elements in a set. Other topologies, such as the discrete topology, do not take into account the ordering of elements and simply define open sets as any subset of the set.

Similar threads

Back
Top