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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").

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  • $\begingroup$ It seems that this condition is equivalent to: the closure of an $R$-saturated set is $R$-saturated. $\endgroup$
    – Taras Banakh
    Commented Sep 1, 2016 at 14:37
  • $\begingroup$ This property is related to the rotundness (of quasi-uniformities) considered in dx.doi.org/10.1016/j.topol.2015.12.010 $\endgroup$
    – Taras Banakh
    Commented Sep 1, 2016 at 14:43
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    $\begingroup$ Rotund quasi-uniformities also are normal quasi-uniformities, see arxiv.org/abs/1410.1504 $\endgroup$
    – Taras Banakh
    Commented Sep 1, 2016 at 14:46
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    $\begingroup$ Does adh denote closure? $\endgroup$
    – Emil Jeřábek
    Commented Apr 12, 2022 at 12:49
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    $\begingroup$ I think "adh" is short for "adherence", which I believe is indeed the same as closure but mostly used in French. $\endgroup$
    – François G. Dorais
    Commented Apr 12, 2022 at 14:43

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