Find a closed interval topology

In summary, we are trying to prove that an ordered set X, where every closed interval is compact, has the least upper bound property. This means that every nonempty subset of X that is bounded from above has a least upper bound. To do this, we consider a subset A of X that is bounded from above and try to find a closed interval from it. We define C as the closure of this interval and show that if C is equal to the interval, it is compact, which leads to a contradiction. Thus, C must have at least one element more than the interval.
  • #1
g1990
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Homework Statement


Let X be an ordered set where every closed interval is compact. Prove that X has the least upper bound property.


Homework Equations


X having the least upper bound property means that every nonempty subset that is bounded from above has a least upper bound, in other words, an upper bound where any number less that it is not an upper bound.
Compact means that every open cover has a finite subcover.


The Attempt at a Solution


Let A be a subset of X that is bounded from above. I know I should try to find a closed interval from this, but I'm not sure where to get it from.
 
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  • #2


Hello,
I'll try to give an intuitive explanation.

"Let A be a subset of X that is bounded from above."
If A has a greatest element, we're done. If A does not, we can focus on an interval B of the form [...[ with [tex]A \cap B = B[/tex]. Note that B has no greatest element thus there is an open cover that has no finite subcover (is there?).
Define [tex]C = \overline{B}[/tex].
  • If C = B, then B is closed and thus compact (CONTRADICTION! (?))
  • So C has at least an element more than B. What is this element?
 

1. What is a closed interval topology?

A closed interval topology is a type of topology that is defined on a set of real numbers. It is created by taking all the closed intervals in the set and treating them as open sets. This results in a topology that allows for continuity and connectedness between points on the set.

2. How is a closed interval topology different from other topologies?

A closed interval topology is different from other topologies because it specifically focuses on closed intervals, whereas other topologies may include open sets, closed sets, or a combination of both. Additionally, a closed interval topology is commonly used in the study of real analysis, while other topologies may be used in different branches of mathematics.

3. What is the importance of studying closed interval topology?

The study of closed interval topology is important because it allows for a deeper understanding of the properties of real numbers and their relationships with one another. Additionally, it has many applications in fields such as physics, economics, and engineering, making it a valuable tool for scientists in various disciplines.

4. How is a closed interval topology related to the concept of continuity?

A closed interval topology is closely related to the concept of continuity because it allows for the definition and study of continuous functions, which are functions that preserve the topological structure of a given set. In other words, if a function is continuous with respect to a closed interval topology, it will preserve the connectedness and continuity of the set it is defined on.

5. Can a closed interval topology be defined on any set of real numbers?

Yes, a closed interval topology can be defined on any set of real numbers. However, it is most commonly used on intervals of the form [a, b] where a and b are real numbers. This is because these intervals have well-defined boundaries and allow for the creation of open sets that are necessary for the closed interval topology to work.

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