Topology intervals on the real line proof

[hlin818]

Homework Statement

a) Let I be a subset of the real line. Prove I is an interval if and only if it contains each point between any two of its points.

b) Let Ia be a collection of intervals on the real line such that the intersection of the collection is nonempty. Show the union of the collection is an interval.

Homework Equations

The Attempt at a Solution

a) An interval is the set (a,b), [a,b], or [a,b). Let I be one of these sets. we want to show that if x and z are in I and x<y<z, then y is in I. Isn't this trivial by how interval is defined?

b) not sure

 

[Dick]

Yes, if I is an interval then the proof of the property is trivial. But you want to show 'iff'. Now you want to show that the betweeness property of I implies I is an interval.

 

[hlin818]

Ah forgot about the other direction. Let some subset of R be I such that I contains each point between any two of its points. Thus if x,z are in I and x<y<z, then y is in I. Let x,z be the endpoints of I. Thus by the betweenness property any y such that x<y<z implies y is in I. So I is an interval.

b)

 

[Dick]

Ah forgot about the other direction. Let some subset of R be I such that I contains each point between any two of its points. Thus if x,z are in I and x<y<z, then y is in I. Let x,z be the endpoints of I. Thus by the betweenness property any y such that x<y<z implies y is in I. So I is an interval.

b)

I is just a set until you show it's an interval. How do you define endpoints of a set?

 

[hlin818]

I is just a set until you show it's an interval. How do you define endpoints of a set?

I suppose by z as the right endpoint if z is greater than or equal to all y in I and likewise for x. So if we define x to be the smallest element in I and z to be the largest, by the betweenness property we have that I contains every point in between x and z, i.e. for any x<y<z, y is in I. Doesnt that mean I is an interval?

 

[Dick]

I suppose by z as the right endpoint if z is greater than or equal to all y in I and likewise for x. So if we define x to be the smallest element in I and z to be the largest, by the betweenness property we have that I contains every point in between x and z, i.e. for any x<y<z, y is in I. Doesnt that mean I is an interval?

Sure it does, but you have to spell all that out in a proof. There's also the possibility that I may not have a smallest or largest element which means you should include the possibility of infinite intervals. I'm not saying the proof isn't easy. You just have to be more explicit about the argument.