A function $\varphi : X \rightarrow Y$ between two topological spaces is continuous if and only if $\varphi(\,\overline{A}\,) \subset  \overline{\varphi(A)}$ for all $A \subset X$.

 This property can be extended to equivalence relations as follows.
 Given an equivalence relation $R$ on $X$ and $A \subset X$, we denote by $R(A)$ the set of points equivalent to points in A. We can look at the property of $R$ given by 

$$R\Bigl(\overline{A}\Bigr) \subset \overline{R(A)} \ \ for\  \ all\ \ A ⊂ X.$$

 Is there a name for that property? Is it studied somewhere? Can it be formulated in term of the continuity of some function associated To $R$?


I encountered this property while studying the equivalence relation associated to a foliation ("being on the same leaf").