An Interval that is both open and closed.

In summary, an interval that is both open and closed must contain all of its boundary points and none of its boundary points. This is only possible if the set is either equal to R or is the empty set. Therefore, I=R or I is the empty set.
  • #1
heshbon
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Homework Statement


Let I C R be an interval which is open and closed at the same time. Prove that I=R or I is the empty set.


Homework Equations





The Attempt at a Solution



I'm looking more for a outline structure for the solution. I have made assumptions that I is not equal to R seeking a contradition, but i don't know what further assumptions to make.
 
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  • #2
A closed set is one that contains all of its boundary points. An open set is one that contains none of its boundary points.

In order to be both open and closed, a set must contain all of its boundary points and none of its boundary points!

If "all boundary points" is the same as "no boundary points", what can you say about the boundary points of the set?
 
  • #3
Well, the boundry point can't be real.., maybe i can say that the set is unbounded
 
  • #4
If a set has NO boundary points the "all boundary points" is the same as "no boundary points". A set is both open and closed if and only if it has no boundary points. What are the boundary points of the empty set? What are the boundary points of R?

(A point, p, is a boundary point of set, S, if and only if every neighborhood of p contains points of both A and the complement of S.)
 

Related to An Interval that is both open and closed.

1. What is an interval that is both open and closed?

An interval that is both open and closed is a set of real numbers that includes its endpoints and all the numbers in between, but does not include any numbers outside of the interval.

2. How is an interval that is both open and closed represented?

An interval that is both open and closed is typically represented using square brackets [ ] for closed intervals and parentheses ( ) for open intervals. For an interval that is both open and closed, the notation used is [ ).

3. What is the difference between an open interval and a closed interval?

An open interval does not include its endpoints, while a closed interval includes its endpoints. For example, the open interval (0, 1) includes all real numbers between 0 and 1, but does not include 0 or 1. The closed interval [0, 1] includes all real numbers between 0 and 1, including 0 and 1.

4. Why is an interval that is both open and closed important in mathematics?

An interval that is both open and closed is important in mathematics because it allows for more precise and accurate calculations and descriptions of sets of real numbers. It also helps to avoid ambiguity in mathematical statements and proofs.

5. Can an interval that is both open and closed be infinite?

Yes, an interval that is both open and closed can be infinite. For example, the interval [0, ∞) includes all real numbers greater than or equal to 0, while the interval (-∞, ∞) includes all real numbers.

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