Metric spaces and closed sets

Hi, I need some help with the following question:

Let be a metric space and . Define the set for as

Prove the following statements:

(1)

(2)

(3)

(4)

(5) Does there exist an example wherefore

Proofs:

(1) Suppose thus by definition (with ) therefore

Can someone give some hints to prove the other 4 statements, because I'm stuck there.

Re: Metric spaces and closed sets

Quote:

Originally Posted by

**Siron** Hi, I need some help with the following question:

Let

be a metric space and

. Define the set

for

as

Prove the following statements:

(1)

(2)

(3)

(4)

(5) Does there exist an example wherefore

Proofs:

(1) Suppose

thus by definition

(with

) therefore

From #1 you already have half of #2. So show that .

In #3 I would show the complement is open.

#4 should fall right from applying the definition of boundary point.

Re: Metric spaces and closed sets

To prove:

Proof:

Suppose

thus which means

and therefore

, write

But , therefore we can choose an element with

and so

What do you think about this proof?

Re: Metric spaces and closed sets

Re: Metric spaces and closed sets

Quote:

Originally Posted by

**Plato** What would it mean if

.

It means that

Quote:

Originally Posted by

**Plato**

I never have seen something like before, what does it mean?

Re: Metric spaces and closed sets