Interior of a Set in a Metric Space: Explained

[mateomy]

"In a metric space M with A as a subset, an interior is the largest open subset contained in A."

That's how I've written this down in my notes along with some more symbolic definition. If the set within M (say, A) is open would/could the interior also be the whole set?


Thanks. Just gettin' ready for my Analysis final.

 

[LCKurtz]

"In a metric space M with A as a subset, an interior is the largest open subset contained in A."

That's how I've written this down in my notes along with some more symbolic definition. If the set within M (say, A) is open would/could the interior also be the whole set?


Thanks. Just gettin' ready for my Analysis final.

You can likely answer this yourself. Think about A = [a,b] versus (a,b) in the reals.

 

[mateomy]

I believe so. Because I know that a subset can in fact be the set itself. And by my own definition "the largest open set contained in A" is another way of saying the largest open subset in A...being A. In the reals, at least. I think my hang up is on the name "interior", is my reasoning correct though?

 

[LCKurtz]

If the set within M (say, A) is open would/could the interior also be the whole set?

I believe so. Because I know that a subset can in fact be the set itself. And by my own definition "the largest open set contained in A" is another way of saying the largest open subset in A...being A. In the reals, at least. I think my hang up is on the name "interior", is my reasoning correct though?

(a,b) is the largest open set contained in (a,b) so the answer to your question is yes.

 

[mateomy]

Awesome. At least I know if this question is on the test (yeah right), I'll be fine.

Thanks.