The following question is from Fred H. Croom's book "Principles of Topology"
Let $(X,d)$ be a metric space and $x_1,x_2$ distinct points of $X$. Prove that there are disjoint open sets $O_1$ and $O_2$ containing $x_1$ and $x_2$, respectively.
How would I approach this problem?
My attempt thus far was to show that $x_1$ and $x_2$ respectively have infinitely many elements in their own neighborhoods: $O_1 $ and $O_2$.
Once you reach a small enough neighborhood for both distinct points, the intersection between the two open sets would be disjoint. To strengthen the claim, I wanted to show that the distance between the two sets would at one point be greater than $0$, proving they are disjoint.
Am I approaching this the right way? If not, how would I properly prove this?
I want to thank you for taking the time to read this question. I greatly appreciate any assistance you provide.