I am in a introductory topology class and our class was given a problem to think about and I am having trouble approaching it. He gave us a set X = $\mathbb R^n$ and a base:
$\beta$ = {all open balls $\ B_{\delta,n}(x) | \forall x\in \Bbb R^n, \delta > 0 $}. The question is to show that this is in fact a basis. It is pretty trivial to show that given any value in $\mathbb R^n$ we can form an open ball in the space. I am not sure how to approach the problem with two open balls intersecting and indicating that we can place any open ball within the intersection. The professor gave a hint about using the triangle inequality but I am otherwise kind of stuck. This is not a homework question just a practice question to try. Thanks for any help!
PS: Sorry about the weird formatting.