An Interval that is both open and closed.

[heshbon]

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.

 

[HallsofIvy]

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?

 

[heshbon]

Well, the boundry point can't be real.., maybe i can say that the set is unbounded

 

[HallsofIvy]

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.)