Definition of Adherent Point

Question

I am using the following definition:

(X,d) a metric space, A is a subset of X, for any x in X, it is an adherent point of A if its neighborhood Br(x) contains an element in A for all r>0.

For one of my exercise, it would be really useful if I can use another definition of adherent point:

(X,d) a metric space, A is a subset of X, for any x in X, it is an adherent point of A if every open set in X containing x contains an element in A.

How can I show these two definitions are equivalent?

Answer 1

Those are equivalent in metric spaces: for every open $O$ that contains $x$, there is some $r>0$ such that $B(x,r) \subseteq O$ ($O$ is a union of balls, or the definition of open sets in a metric space). So if every open ball around $x$ intersects $A$, so does every open set containing $x$.

The reverse also holds, as open balls are themselves in particular open sets in the topology.