Difference between closed set and bounded set

1. Sep 10, 2007

kthouz

The way they use the terms:"closed set" and "bounded set" make me thinking that a closed set is different from a bounded set but i can not figure out how to prove that. So can some body show me clearly the difference between those two terms?

2. Sep 10, 2007

matt grime

Yes. Write down a closed set that is not bounded, and a bounded set that is not closed. The two things aren't really related, so I wonder what your definitions of them are.

3. Sep 10, 2007

mathman

Some examples (usual topology for real line):

Closed but not bounded - entire line

Bounded but not closed - 0<x<1

This should give you some idea for proofs.

4. Sep 13, 2007

kthouz

Now am getting the point but the question again risesow can a set be closed without being bounded?

5. Sep 13, 2007

JonF

It seems like you are trying to think of this too intuitively. What does your definition say a closed set is? What does your definition say a bound set is?

You asked how a set can be closed without being bound, but mathman just showed you. By being the entire space.

6. Sep 13, 2007

CRGreathouse

Another example would be [0, 1] U [5, 6] U [10, 11] U [15, 16] U ... .

7. Sep 13, 2007

kthouz

ok now i understand with this example. Thank you

8. Sep 23, 2008

hamchaley

In fact it is one sided relation , that every closed set is bounded but converse is not necessarily true , example is above cited [0,1] and [0,1) two are bounded but second one is not closed

9. Sep 23, 2008

NoMoreExams

I thought someone just gave an example of closed but not bounded, the entire line.

10. Oct 21, 2010

RobNichol

I might be wrong, but I think this is an incorrect example.. If you have a collection of closed sets, I believe only their intersection would be closed, not the union.

11. Oct 21, 2010

Office_Shredder

Staff Emeritus
The union of closed sets is not always closed, but it can be closed. In particular, the one Greathouse posted is closed (you can just check this with the definition of closed)

12. Jun 10, 2012

cameroncope

According to the definitions in my analysis course:

The real line is closed because its complement, the empty set, is open.

Obviously the real line is not bounded because there is no upper bound and no lower bound.

So the real line is an example of a closed, unbounded set from that perspective.

13. Jun 10, 2012

micromass

Staff Emeritus
This thread is 5 years old.