If a property holds for arbitrary compact set in a metric space, does it also holds for the metric space?

Question

Suppose a metric space $(X, d).$ Further suppose that a property $A$ holds for arbitrary compact subset of $X.$ Does the property $A$ also hold for $X$?

Context

I hoped for some general theorems of the above kind. I am not expert in mathematical analysis so I hoped that there exists a class of properties that also hold for the space when they hold for its compact subsets.

Answer 1

If $X$ is a compact metric space, then $X$ itself is a compact subset of $X$, so the property $A$ must hold for $X$.

If $X$ is not compact, then let $A$ be the property that a subset of $X$ is compact. Then $A$ holds for any compact subset of $X$, but $A$ does not hold for $X$ itself.

Answer 2

No; there are , e.g., fixed point theorems that hold for compact subspaces, but not for the whole space. Or, you may have subspaces, like $\mathbb Q$ of the Reals that are not complete under the standard Euclidean metric, but the Reals themselves are complete.