Recently I encountered a mysterious term named "metric $2$-sphere". Does it have anything to do with a metric space? If it is typically used to refer to a sphere in any metric space, why should I mention the number $2$? Is it like the superscript $n$ in the $n$-sphere $\mathbb{S}^n$?
As to a "topological sphere", this is probably a concept that is easier to understand. After I googled this term, it seems to represent a topological space that is homeomorphic to a sphere in a Euclidean space w.r.t. the subspace topology.
Then, what is a metric $2$-sphere supposed to mean? Thank you.