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?

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.

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.

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