Difference between closed set and bounded set

[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?

 

[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.

 

[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.

 

[kthouz]

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

 

[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.

 

[CRGreathouse]

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

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

 

[kthouz]

ok now i understand with this example. Thank you

 

[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

 

[NoMoreExams]

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

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

 

[RobNichol]

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

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.

 

[Office_Shredder]

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)

 

[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.

 

[micromass]

This thread is 5 years old.