Question

3. Prove that any open ball in a metric space (X, d) is open and any
closed ball is closed.

Answer 1

Let \(X\) be a metric space and let \(G=B(x_0;r)\) be an open ball in \(X\).
Fix any \(a\in G\).
Then \(d(x_0,a)<r\), thus \(\delta = r - d(x_0,a)>0.\)
Let \(x\in B(a;\delta)\),
then \(d(x,a)< \delta = r-d(x_0,a)\).
It follows that \(d(x,x_0)\le d(x,a)+d(a,x_0)<r-d(x_0,a)+d(x_0,a)=r\) as desired, and thus \(B(x_0,r)\) is an open set.

Answer 2

Let me know if this makes sense then we can move on to the closed ball problem.

Answer 3

yes.. i think that makes sense..let me try it first..