Correct definition of locally uniform convergence

Question

We have a sequence of complex functions $f_n\colon D \subseteq \mathbb{C} \to \mathbb{C}$ where $D$ is a generic set.

What is the correct definition of locally uniform convergence of the sequence $(f_n)$ to the function $f\colon D \to \mathbb{C}$ in the set $D$?

I have found (as possible definition) the following ones:

1. $\forall \,z \in D \quad \exists \,B_r(z)=\{w \in \mathbb{C} \mid |w-z|<r\} \mid (f_n)$ uniformly converges to $f$ on $B_r(z) \cap D$;

2. $\forall \,K \subseteq D \mid K$ is compact, we have that $(f_n)$ uniformly converges to $f$ on $K$;

3. For every closed disk $B$ contained in $D$ we have that $(f_n)$ uniformly converges to $f$ on $B$.

I have found that the three definitions are equivalent if $D$ is an open set. Is this true?

Thank you!