Is it a base for a Topology in R^N

Hi there

Prove this statement or the opposite:

Consider the set . Define .

The collection is a base for a topology on .

Is this true or not? Why? I'm sorry but I don't understand a lot after reading this multiple times?? How do I show that it is a base of a topology and which one?

Could maybe please someone give me a hint?

Regards

Re: Is it a base for a Topology in R^N

What definition of "base" for a topology are you using?

Re: Is it a base for a Topology in R^N

Quote:

Originally Posted by

**huberscher** Prove this statement or the opposite:

Consider the set

. Define

.

The collection

is a base for a topology on

.

It is fairly clear that .

Can you show that

Re: Is it a base for a Topology in R^N

Thank you.

So I have to differ between the disjoint and the non-disjoint case, don't I ?

disjoint case:

For r=0 U(0)= and so it is true that .

Hmm, I am not convinced of this argument because one could always head this statement for S, T arbitrary. There was always ?

non-disjoint case:

Then

fullfills the condition above.

So since ??

So since ??

What about the other parameters used by "min". They make sure that the whole Ball is contained in . How can I show formally that they are needed and make sure that the statement is true?

So the topology is

Obviously the family of balls

where covers

What do you think?

Regards