# Prove: Interior is an open set

• Feb 20th 2010, 12:45 AM
kingwinner
Prove: Interior is an open set
Let (X,d) be a metric space and let F be a subset of X. B(r,x) is the open ball of radius r about x.
Definition: The interior of F, int(F), is the set of all x E F such that there is an r > 0 such that B(r,x) is contained in F.
Definition: Let D be a subset of X. By definition, D is open iff for all a E D, there exists r>0 such that B(r,a) is contained in D.

Using these definitions, prove that int(F) is an open set.

============================

Attempt:
Let x E int(F), then there exists r>0 such that B(r,x) is contained in F.
We need to prove that there exists some r' >0 s.t. B(r',x) is contained in int(F). How can we prove this?

Any help is greatly appreciated!
• Feb 20th 2010, 02:08 AM
Opalg
Quote:

Originally Posted by kingwinner
Let (X,d) be a metric space and let F be a subset of X. B(r,x) is the open ball of radius r about x.
Definition: The interior of F, int(F), is the set of all x E F such that there is an r > 0 such that B(r,x) is contained in F.
Definition: Let D be a subset of X. By definition, D is open iff for all a E D, there exists r>0 such that B(r,a) is contained in D.

Using these definitions, prove that int(F) is an open set.

============================

Attempt:
Let x E int(F), then there exists r>0 such that B(r,x) is contained in F.
We need to prove that there exists some r' >0 s.t. B(r',x) is contained in int(F). How can we prove this?

Any help is greatly appreciated!

As a very brief hint, try taking r' = r/2 and using the triangle inequality.
• Feb 20th 2010, 06:00 AM
Drexel28
Quote:

Originally Posted by kingwinner
Let (X,d) be a metric space and let F be a subset of X. B(r,x) is the open ball of radius r about x.
Definition: The interior of F, int(F), is the set of all x E F such that there is an r > 0 such that B(r,x) is contained in F.
Definition: Let D be a subset of X. By definition, D is open iff for all a E D, there exists r>0 such that B(r,a) is contained in D.

Using these definitions, prove that int(F) is an open set.

============================

Attempt:
Let x E int(F), then there exists r>0 such that B(r,x) is contained in F.
We need to prove that there exists some r' >0 s.t. B(r',x) is contained in int(F). How can we prove this?

Any help is greatly appreciated!

That is a strange definition for the interior. The way that I have most often seen it defined as is the union of all open subsets of the set.
• Feb 20th 2010, 12:39 PM
kingwinner
Quote:

Originally Posted by Opalg
As a very brief hint, try taking r' = r/2 and using the triangle inequality.

So you mean B(r/2,x) is contained in int(F)? How can we prove this? What is the intermediate point in the triangle inequality?

thanks.
• Feb 20th 2010, 01:02 PM
Opalg
Quote:

Originally Posted by kingwinner
So you mean B(r/2,x) is contained in int(F)? How can we prove this? What is the intermediate point in the triangle inequality?

If $y\in B(r/2,x)$ then $B(r/2,y)\subseteq B(r,x)\subseteq F$, because $d(z,y) and $d(y,x) together imply that $d(z,x), by the triangle inequality. Therefore $B(r/2,x)\subseteq \text{int}(F)$.