# Closed Set Requirements

• Dec 22nd 2011, 10:35 AM
bugatti79
Closed Set Requirements
Folks,

Generally speaking, what must I use to show that a certain set of elements is a closed subset of some other larger set for a certain norm?

Thanks
• Dec 22nd 2011, 10:49 AM
Plato
Re: Closed Set Requirements
Quote:

Originally Posted by bugatti79
Generally speaking, what must I use to show that a certain set of elements is a closed subset of some other larger set for a certain norm?

Is this a question about a general topological space?
You used the word norm. Does that imply that this is a metric space?

Or is the some other setting altogether for closed subsets of elements?
• Dec 22nd 2011, 10:54 AM
bugatti79
Re: Closed Set Requirements
Quote:

Originally Posted by Plato
Is this a question about a general topological space?
You used the word norm. Does that imply that this is a metric space?

Or is the some other setting altogether for closed subsets of elements?

Yes, I am referring to a metric space....... (I suspect the question is too vague for a reasonable answer.)

Thanks Plato.
• Dec 22nd 2011, 10:59 AM
Plato
Re: Closed Set Requirements
Quote:

Originally Posted by bugatti79
Yes, I am referring to a metric space....... (I suspect the question is too vague for a reasonable answer

Two general approaches come to mind at once.
First, show that the set contains all of its limit points.

Second, equivalently show that its complement is open.
• Dec 22nd 2011, 11:26 AM
bugatti79
Re: Closed Set Requirements
Quote:

Originally Posted by Plato
Two general approaches come to mind at once.
First, show that the set contains all of its limit points.

Second, equivalently show that its complement is open.

For this example...is the set

$\displaystyle \{z \in C[a,b]: u\le z(s)\le v \forall s \in [a,b]\}$

a closed subset in C[a,b] with the integral norm?

How would i tackle this?

thanks