Question about open extensions of sets in metric spaces.

Let be an arbitrary metric space. For every and , we define the open -extension of as the set

,

where .

It seems intuitive that, given and , we should have that . However, I haven't been able to even begin to construct a rigorous argument to prove it. Any hint or proposed direction would be greatly appreciated (or counterexample in the event that it is false).

Re: Question about open extensions of sets in metric spaces.

Quote:

Originally Posted by

**RaisinBread** Let

be an arbitrary metric space. For every

and

, we define the open

-extension of

as the set

,

where

.

It seems intuitive that, given

and

, we should have that

. However, I haven't been able to even begin to construct a rigorous argument to prove it. Any hint or proposed direction would be greatly appreciated (or counterexample in the event that it is false).

When you say that you are unable "to construct a rigorous argument", does that mean that you cannot even start?

Can you show that

Before I work on this, please post what you have been able to do.