Let $(X, d)$ be a metric space. Is it true that if one singleton set $\{x\}$ is open, all singletons sets in $X$ must be open?
Motivation: The only example of a metric I know where singletons are open is the discrete metric, but in this example all the singletons are open. For example, if $X=\{a, b\}$, what I'm trying to find is a metric such that the only open sets are $\{{\emptyset}, \{a\}, \{a, b\}\}$.
On
As Dr. Santos points out in the previous answer, it is not generally true that in a metric space, if one singleton is open they all are.
With respect to your motivation for the question, I would say that you need to ask a slightly different question: "Are singletons always open in a finite metric space?"
The answer to this question should come to you if think about how the basis is defined on a metric topology, coupled with the fact that in a finite metric space, there will be a minimal possible distance between two distinct points.
No, it is not true. Take $X=\{0\}\cup[1,2]$, endowed with the usual metric. The only open singleton is $\{0\}$.