A special kind of metric-spaces

Question

Is there a special name for those metric-spaces or topological spaces in which every non-empty open set is uncountable ?

Answer 1

Suggested name „nowhere locally countable“ makes perfect sense since it matches the pattern $\text{nowhere-}φ(X) \iff (∀\text{ open }U ⊆ X): ¬φ(U)$. Note that this is equivalent to shorter „nowhere countable“. This also works for nowhere dense subsets with $A ⊆ X$ pattern $\text{nowhere-}φ(A, X) \iff (∀\text{ open }U ⊆ X): ¬φ(A ∩ U, U)$.

There is also a cardinal invariant of topological spaces $Δ(X) = \min\{\lvert U\rvert: U ⊆ X \text{ open}\}$ called dispersion character and used in resolvability theory. So your condition is equivalent to $Δ(X) > ω$.