A function $\varphi : X \rightarrow Y$ between two topological spaces is continuous if and only if $\varphi(adh A) \subset adh (\varphi A)$ 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").

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").

A function φ : X → Y between two topological spaces is continuous if and only if φ(adh A) ⊂ adh (φ A) for all A ⊂ X.

This property can be extended to equivalence relations as follows. Given an equivalence relation R on X and A ⊂ 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").

A function $\varphi : X \rightarrow Y$ between two topological spaces is continuous if and only if $\varphi(adh A) \subset adh (\varphi A)$ 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").